COEFFICIENT SYSTEMS AND SUPERSINGULAR … · In general we use the results of Jeyakumar [10], where...

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COEFFICIENT SYSTEMS ANDSUPERSINGULAR

REPRESENTATIONS OF GL2(F )

Vytautas Paskunas

Societe Mathematique de France 2004Publie avec le concours du Centre National de la Recherche Scientifique

V. PaskunasFakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld,Germany.E-mail : [email protected]

2000Mathematics Subject Classification. — 22E50.Key words and phrases. — Supersingular, mod p-representations.

COEFFICIENT SYSTEMS AND SUPERSINGULARREPRESENTATIONS OF GL2(F )

Vytautas Paskunas

Abstract. — Let F be a non-Archimedean local field with the residual characteristic p.We construct a “good” number of smooth irreducible Fp-representations of GL2(F ),which are supersingular in the sense of Barthel and Livné. If F = Qp then resultsof Breuil imply that our construction gives all the supersingular representations upto the twist by an unramified quasi-character. We conjecture that this is true for anarbitrary F .

Résumé (Systèmes de coefficients et représentations supersingulières de GL2(F ))Soit F un corps local non archimédien de caractéristique résiduelle p. Nous construi-

sons le « bon » nombre de Fp-représentations lisses et irréductibles de GL2(F ) qui sontsupersingulières au sens de Barthel et Livné. Si F = Qp, les résultats de Breuil im-pliquent alors que notre construction donne toutes les représentations supersingulièresà la torsion près par un quasi-caractère non ramifié. Nous conjecturons que ceci restevrai pour F quelconque.

c! Mémoires de la Société Mathématique de France 99, SMF 2004

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2. Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1. Supersingular modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2. Restriction to HK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3. Irreducible representations of GL2(Fq) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1. Carter and Lusztig theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2. Alternative description of irreducible representations . . . . . . . . . . . . . . . . . . . . 21

4. Principal indecomposable representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1. The case q = p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2. The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5. Coe!cient systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2. Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.3. Some computations of H0(X,V) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.4. Constant functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.5. Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6. Supersingular representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.1. Coe!cient systems V! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2. Injective envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.3. Admissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.4. Coe!cient systems I! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.5. Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

CHAPTER 1

INTRODUCTION

Recently Breuil in [4] has determined the isomorphism classes of the irreduciblesmooth Fp-representations of GL2(Qp). This allowed him to define a “correspon-dance semi-simple modulo p pour GL2(Qp)”. Under this correspondence the iso-morphism classes of irreducible smooth 2-dimensional Fp-representations of the Weilgroup of Qp are in bijection with the isomorphism classes of “supersingular” irre-ducible smooth Fp-representations of GL2(Qp). Moreover, it is conjecturally relatedto a p-adic correspondence in [5]. The term“supersingular”was coined by Barthel andLivne. Roughly speaking a supersingular representation is the Fp-analogue of a su-percuspidal representation over C, see Definition 1.1.1. Let F be a non-Archimedeanlocal field, with a residue class field Fq of the characteristic p. All the irreduciblesmooth Fp-representations of G = GL2(F ), which are not supersingular, have beendetermined by Barthel and Livne in [2] and [1], and also by Vigneras in [18], with norestrictions on F . However, if F != Qp then the method of Breuil fails and relativelylittle is known about the supersingular representations of G.

This paper is an attempt to shed some light on this question. We fix a uni-formiser !F of F and we construct q(q " 1)/2 pairwise non-isomorphic, irreducible,supersingular, admissible (in the usual smooth sense) representations of G, whichadmit a central character, such that !F acts trivially. If F = Qp then using theresults of Breuil we may show that our construction gives all the supersingular rep-resentations up to a twist by an unramified quasi-character. We conjecture that thisis true for arbitrary F . If " is an irreducible smooth Fp-representation of the Weilgroup WF of F , then the wild inertia subgroup of WF acts trivially on ", since it ispro-p and normal in WF . This implies that there are only q(q " 1)/2 isomorphismclasses of irreducible smooth 2-dimensional Fp-representations " of the Weil group ofF such that (det ")(Fr) = 1. Here, Fr is the Frobenius automorphism correspondingto !F via the local class field theory. So the conjecture would be true if there was aLanglands type of correspondence.

2 CHAPTER 1. INTRODUCTION

The starting point in this theory is that every pro-p group acting smoothly onan Fp-vector space has a non-zero invariant vector. Let I1 be the unique maximalpro-p subgroup of the standard Iwahori subgroup I of G. Given a smooth represen-tation # of G the Hecke algebra H = EndG(c-IndG

I1 1) acts on the I1-invariants #I1 .It is expected that this functor induces a bijection between the irreducible smoothrepresentations of G and the irreducible modules of H. This happens if F = Qp.Moreover, if F is arbitrary and # is an irreducible smooth representation of G, whichis not supersingular, then #I1 is an irreducible H-module. All the irreducible modulesof H that do not arise this way are called supersingular. They have been determinedby Vigneras and we give a list of them in the Definition 2.1.2. There are q(q " 1)/2isomorphism classes of irreducible supersingular modules of H up to a twist by anunramified quasi-character.

Given a supersingular module M of H we construct two G-equivariant coe!cientsystems V and I on the Bruhat-Tits tree X of PGL2(F ) and a morphism of G-equivariant coe!cient systems between them. Once we pass to the 0-th homology,this induces a homomorphism of G-representations. We show that the image of thishomomorphism

# = Im(H0(X,V) "# H0(X, I))

is a smooth irreducible representation of G, which is supersingular, since #I1 con-tains a supersingular module M . Moreover, we show that two non-isomorphic irre-ducible supersingular modules give rise to non-isomorphic representations. However,the question of determining all smooth irreducible representations # of G, such that#I1 contains M , remains open.

We will describe the contents of this paper in more detail. In Section 2 we recallthe algebra structure of H and the definition of supersingular modules.

Sections 3 and 4 deal with some aspects of the Fp-representation theory of " =GL2(Fq). In Section 3 we give two di#erent descriptions of the irreducible Fp-representations of ", one of them due to Carter and Lusztig [7] and the other one dueto Brauer and Nesbitt [3], and a dictionary between them. Let U be the subgroupof unipotent upper-triangular matrices in ", then U is a p-Sylow subgroup of ". If "is a representation of ", then the Hecke algebra H! = End!(Ind!

U 1) acts on the U -invariants "U . This functor induces a bijection between the irreducible representationsof " and the irreducible right modules of H!.

Every representation " of " has an injective envelope $ : " %# inj ". By this wemean, a representation inj " of " and an injection $, such that inj " is an injective objectin the category of Fp-representations of " and every non-zero "-invariant subspace ofinj " intersects $(") non-trivially. Injective envelopes are unique up to isomorphism.In Section 4 we determine the H!-module structure of (inj ")U , for an irreduciblerepresentation " of ". This is important to us, so we give two ways of doing it. If


CHAPTER 1. INTRODUCTION 3

p = q then the dimension of (inj ")U is small and this enables us to give an elemen-tary argument. In general we use the results of Jeyakumar [10], where he describesexplicitly injective envelopes of irreducible representations of SL2(Fq).

Let oF be the ring of integers of F , let K = GL2(oF ). The reduction modulothe prime ideal of oF induces a surjection K # ", let K1 be the kernel of this map.The Hecke algebra HK = EndK(IndK

I1 1) is naturally a subalgebra of H. Let M be asupersingular module of H, then the restriction of M to HK is isomorphic to a directsum of two irreducible modules of HK . Since K/K1

$= " we may identify represen-tations of K on which K1 acts trivially with the representations of ". This inducesan identification HK = H!. Since the irreducible modules of H! are in bijectionwith the irreducible representations of ", there exists a unique representation " = "M

of ", such that " is isomorphic to a direct sum of two irreducible representationsof ", and "U $= M |H! . Let " %# inj " be an injective envelope of " in the category ofFp-representations of ". We consider now both " and inj " as representations of K.We have an exact sequence

0 "# "I1 "# (inj ")I1

of HK-modules. The main result of Section 4 are Propositions 4.1.9 (p = q), Propo-sitions 4.2.37 and 4.2.38 (general case), which say that there exists an action of H,extending the action of HK , on (inj ")I1 , such that the above exact sequence yieldsan exact sequence

(E) 0 "# M "# (inj ")I1

of H-modules. The fact that we can extend the action and obtain (E) implies theexistence of a certain G-equivariant coe!cient system I on the tree X .

The inspiration to use coe!cient systems comes from the works of Schneider andStuhler [13] and [14], where the authors work over the complex numbers, and Ro-nan and Smith [12], where the Fp coe!cient systems are studied for finite Chevalleygroups. We introduce coe!cient systems in Section 5. Let &1 be an edge on X contain-ing a vertex &0. Since, G acts transitively on the vertices of the tree X , the categoryof G-equivariant coe!cient systems is equivalent to a category of diagrams DIAG.The objects of DIAG are triples ("0, "1, '), where "0 is a smooth representation ofK(&0), "1 is a smooth representation of K(&1) and ' is a K(&1)%K(&0)-equivariant ho-momorphism, ' : "1 # "0, where K(&0) and K(&1) are the G-stabilisers of &0 and &1.The proof of equivalence between the two categories is the main result of Section 5.As a corollary we obtain a nice way of passing from “local” to “global” information,see Corollary 5.5.5, and we use this in the construction of I.

More precisely, we start with a supersingular H-module M and find the uniquesmooth representation " = "M of K, such that " is isomorphic to a direct sum of twoirreducible representations of K, and "I1 $= M |HK , as above. We then consider aninjective envelope " %# Inj " of " in the category of smooth Fp-representations of K.

SOCIETE MATHEMATIQUE DE FRANCE 2004


Let &1 be an edge on X fixed by I and let &0 be a vertex fixed by K. We extend theaction of K on Inj " to the action of F!K = K(&0), so that a fixed uniformiser actstrivially. We denote this representation by Y0. Let us assume that we may extendthe action of F!I = K(&1) % K(&0) on Y0|F!I to the action of K(&1). We denote thecorresponding representation of K(&1) by Y1. The triple Y = (Y0, Y1, id) is an objectin a category DIAG, which is equivalent to the category of G-equivariant coe!cientsystems on the tree X , by the main result of Section 5. So Y gives us a G-equivariantcoe!cient system I. Moreover, the restriction maps of I are all isomorphisms. Thisimplies that

H0(X, I)|K $= Inj ".

In particular, we have an injection

" %"# Inj " $= H0(X, I!)|K ,

which gives us an exact sequence of vector spaces

0 "# "I1 "# H0(X, I)I1 .

We show in Subsection 6.4 that using (E) we may extend the action of F!I on Y0|F!I

to the action of K(&1), so that the image of "I1 in H0(X, I)I1 is an H-invariantsubspace, isomorphic to M as an H-module. We let # be the G-invariant subspaceof H0(X, I) generated by the image of ". In Theorem 6.5.2 we prove that # isirreducible and supersingular. We also show that # is the socle of H0(X, I). Thespace H0(X, I)I1 is always finite dimensional, we determine the H-module structurein Proposition 6.4.5. The proofs rely on some general properties of injective envelopes,which we recall in Subsection 6.2. Using injective envelopes we also give a new proofof the criterion for admissibility of a smooth representation of G, which works in avery general context, see Subsection 6.3.

We would like to explain the thinking behind the construction of the coe!cientsystem V in Subsection 6.1. Let # be a smooth representation of G, generated byits I1-invariant vectors. We may associate to # a G-equivariant coe!cient systemF" as follows. Given a simplex & on X , we let U1

# be the maximal normal pro-psubgroup of the G-stabiliser of &. With this notation U1

#1= I1 and U1

#0= K1. We

may consider the coe!cient system of invariants F" = (#U1! )#, where the restriction

maps are inclusions. Since # is generated by its I1-invariants the natural map

H0(X,F") "# #

is surjective. If we are working over the complex numbers then a theorem of Schneiderand Stuhler in [13], says that the above homomorphism is in fact an isomorphism. Ifwe are working over Fp, then H0(X,F") can be much bigger than #.

The construction of V is motivated by the following question. Let M be a su-persingular module of H and suppose that there exists a smooth irreducible Fp-representation # of G such that #I1 $= M . What can be said about the corresponding


1.1. NOTATION 5

coe!cient system F"? It is enough to understand the action of K on #K1 . Thisreduces the question to the representation theory of GL2(Fq). In Corollary 6.1.10 weshow that there exists an injection V %# F" and hence every # as above is a quotientof H0(X,V). We would like to point out that although in most cases we do not knowwhether such # exists, the coe!cient system V is always well defined. Moreover, if #is any non-zero irreducible quotient of H0(X,V), then we show that # is supersingu-lar, since #I1 contains a supersingular H-module M . This implies that H0(X,V) is aquotient of one of the spaces considered by Barthel and Livne in [1]. Corollary 6.1.8implies that at least in some cases the quotient map is an isomorphism. Now the Re-marque 4.2.6 in [4] shows that in general dim H0(X,V)I1 is infinite. The irreduciblerepresentation #, which we construct in this paper, is a quotient of H0(X,V), more-over the space #I1 is finite dimensional. Hence, in contrast to the situation over C, ingeneral H0(X,V) is very far away from being irreducible.

We believe that our construction of irreducible representations will work for othergroups. Our strategy could be applied most directly to the group G = GLN (F ), whereN is a prime number. If N is prime then the maximal open, compact-mod-centresubgroups of G are the G-stabilisers of chambers (simplices of maximal dimension)and vertices in the Bruhat-Tits building of G and if we had the equivalent of (E)then the construction of the coe!cient system I and our proofs would carry through.However, in order to do this one needs to understand the H!-module structure of(inj ")U , (or at least the action of B on (inj ")U , at the cost of not knowing H-modulestructure of H0(X, I)I1), where " is an irreducible Fp-representation of " = GLN (Fq),B is the subgroup of upper-triangular matrices, and U is the subgroup of unipotentupper-triangular matrices of ". This might be quite a di!cult problem, since alreadyfor N = 2 the dimension of (inj ")U can be as big as 2n " 1, if q = pn.

Acknowledgements. — I would like to thank Michael Spiess and Thomas Zink for anumber of useful discussions and for looking after me in general. I would like to thankMarie-France Vigneras for her encouragement and her comments on this work.

1.1. Notation

Let F be a non-Archimedean local field, oF its ring of integers, pF the maximalideal of oF . Let p be the characteristic and let q be the number of elements of theresidue class field of F . We fix a uniformiser !F of F .

Let G = GL2(F ) and K = GL2(oF ). Reduction modulo pF induces a surjectivehomomorphism

red : K "# " = GL2(Fq).

Let K1 be the kernel of red. Let B be the subgroup of " of upper triangular matrices.Then

B = HU



where H is the subgroup of diagonal matrices and U is the subgroup of unipotentmatrices in B. It is of importance, that the order of H is prime to p and U is ap-Sylow subgroup of ". Let I and I1 be the subgroups of K, given by

I = red"1(B), I1 = red"1(U).

Then I is the Iwahori subgroup of G and I1 is the unique maximal pro-p subgroup ofI. Let T be the subgroup of diagonal matrices in K, and let T1 = T % K1 = T % I1.Let N be the normaliser of T in G. We introduce some special elements of N . Let

$ =!

0 1!F 0

", ns =

!0 "11 0

", s =

!0 11 0

".

The images of $ and ns in N/T , generate it as a group. The normaliser N acts on Tby conjugation, and hence it acts on the group of characters of T . This action factorsthrough T , so if w & N/T and ( is a character of T , we will write (w for the character,given by

(w(t) = ((w"1tw), ' t & T.

Let #B be the group of upper-triangular matrices in G, then #B = #T #U where #T is thegroup of diagonal matrices in G and #U is the group of unipotent matrices in #B.

Definition 1.1.1. — Let # be a smooth irreducible Fp-representation of G, suchthat # admits a central character, then # is called supersingular if # is not a subquo-tient of IndG

eB (, for any smooth quasi-character ( : #B # #B/#U $= #T # F!p .

All the representations considered in this paper are over Fp, unless it is statedotherwise.


CHAPTER 2

HECKE ALGEBRA

Lemma 2.0.2. — Let P be a pro-p group and let # be a smooth non-zero representa-tion of P, then the space #P of P-invariants is non-zero.

Proof. — We choose a non-zero vector v in #. Let " = (Pv)Fpbe a subspace of #

generated by P and v. Since the action of P on # is smooth, the stabiliser StabP(v)has finite index in P , hence " is finite dimensional. Let v1, . . . , vd be an Fp basis of ".The group P acts on " and the kernel of this action is given by

Ker " =d$

i=1StabP(vi).

In particular, Ker" is an open subgroup of P . Hence, P/ Ker" is a finite group, whoseorder is a power of p. Now,

"P = "P/ Ker $ != 0

since P/ Ker" is a finite p-group, see [15], §8, Proposition 26.

Let # be a smooth representation of G, then

#I1 $= HomI1(1, #) $= HomG(c-IndGI1 1, #)

by Frobenius reciprocity. Let H be the Hecke algebra

H = EndG(c-IndGI1 1)

then via the above isomorphism #I1 becomes naturally a right H-module. We obtaina functor

RepG "# Mod -H, # *"# #I1 ,

where RepG is a category of smooth Fp-representations of G and Mod -H is thecategory of right H-modules. Since I1 is an open pro-p subgroup of G, Lemma 2.0.2implies that #I1 = 0 if and only if # = 0. This functor is our basic tool. The algebrastructure of H is well understood, in a general context of split reductive groups over F ,see [17]. We recall some of the results below. Since we deal only with GL2 we canbe very explicit. Our notation follows [7], where finite groups with split BN -pair aretreated.

8 CHAPTER 2. HECKE ALGEBRA

Definition 2.0.3. — Let g & G and f & c-IndGI1 1 we define Tg & H by

(Tgf)(I1g1) =%

I1g2#I1g"1I1g1

f(I1g2).

Lemma 2.0.4. — We may write G as a disjoint union

G =&

n$N/T1

I1nI1

of double cosets.

Proof. — This follows from the Iwahori decomposition.

It is immediate that the definition of Tg depends only on the double coset I1gI1.The Lemma above implies that it is enough to consider Tn, where n & N is a repre-sentative of a coset in N/T1.

Definition 2.0.5. — Let ) & c-IndGI1 1 be the unique function such that

Supp) = I1 and )(u) = 1, 'u & I1.

Lemma 2.0.6

(i) The function ) generates c-IndGI1 1 as a G-representation.

(ii) SuppTn) = I1nI1 and (Tn))(g) = 1, for every g & I1nI1. In particular,

Tn) =%

u$I1/(I1%n"1I1n)

un"1).

(iii) The set {Tn) : n & N/T1} is an Fp-basis of (c-IndGI1 1)I1 .

(iv) The set {Tn : n & N/T1} is an Fp-basis of H.

Proof. — Let g & G, then Supp(g"1)) = I1g and (g"1))(I1g) = 1. Part (i) followsimmediately.

Let f & c-IndGI1 1, then by examining the definition of Tn, one obtains that

Supp(Tnf) + I1n Supp f . Hence, Supp(Tn)) + I1nI1. Since Tn is a G-equivarianthomomorphism and I1 acts trivially on ), it is enough to prove that (Tn))(n) = 1.Since Supp ) = I1, it is immediate from Definition 2.0.3 that (Tn))(I1n) = )(I1) = 1.The last part follows from decomposing I1nI1 into right cosets and applying theargument used in Part (i).

Let n, n& & N , and suppose that nT1 != n&T1, then Lemma 2.0.4 implies thatI1nI1 != I1n&I1. By Part (ii) the functions Tn) and Tn#) have disjoint support. Thisimplies that the set {Tn) : n & N/T1} is linearly independent. Any f & (c-IndG

I1 1)I1 ,is constant on the double cosets I1nI1, for n & N , and since Supp f is compact, f issupported only on finitely many such, hence Lemma 2.0.4 and Part (ii) imply that{Tn) : n & N/T1} is also a spanning set. Hence we get Part (iii).

Let * & H, Part (i) implies that * = 0 if and only if *()) = 0. This observationcoupled with Part (iii) implies Part (iv).


CHAPTER 2. HECKE ALGEBRA 9

Corollary 2.0.7. — Let # be a smooth representation of G and let v & #I1 , thenthe action of Tn on #I1 is given by

vTn =%

u$I1/(I1%n"1I1n)

un"1v.

Proof. — The isomorphism HomG(c-IndGI1 1, #) $= #I1 is given explicitly by * *# *()).

Let * be the unique G-invariant homomorphism, such that *()) = v, then

vTns = (* , Tns)()) = *(Tns)) = *

! %

u$I1/(I1%n"1I1n)

un"1)

".

The last equality follows from Lemma 2.0.6 (ii). Since, * is G-invariant, we obtainthe Lemma.

Lemma 2.0.8. — Let n&, n & N and suppose that n normalises I1, then

Tn#Tn = Tn#n, TnTn# = Tnn# .

Proof. — Lemma 2.0.6 (i) implies that it is enough to show that the homomorphismsmap ) to the same function. Let f & c-IndG

I1 1 then since n normalises I1 we have(Tn(f))(g) = f(ng) and Tn) = n"1). Now the Lemma follows from Lemma 2.0.6(ii).

Let t & T and let h be the image of t in H , via T/T1$= H , we will write Th for the

homomorphism Tt.

Definition 2.0.9. — Let ( : H # F!p be a character, we define

e% =1|H |

%

h$H

((h)Th.

Let)% = e%),

then )% is the unique function in c-IndGI1 1 such that

Supp)% = I, )%(g) = ((gI1), ' g & I,

via the isomorphism I/I1$= H .

Lemma 2.0.10

(i) e2% = e% and e%e%# = 0, if ( != (&.

(ii) id ='

% e%, where the sum is taken over all characters ( : H # F!p .

(iii) e%(c-IndGI1 1) $= c-IndG

I (.

Proof. — We note that H is abelian and the order of H is prime to p. Parts (i) and(ii) follow from the orthogonality relations of characters. Lemma 2.0.6 (i) implies thate%(c-IndG

I1 1) is generated by )% and this implies Part (iii).



Corollary 2.0.11. — Let # be a smooth representation of G, then I acts on (#I1)e%

by a character (. Moreover,#I1 $= -%(#I1)e%.

Proof. — The group I acts on #I1 . Since I1 acts trivially and I/I1$= H , which

is abelian and of order prime to p, the space #I1 decomposes into one dimensional Iinvariant subspaces. Corollary 2.0.7 implies that e% cuts out the (-isotypical subspace.The last part follows from Lemma 2.0.10 (ii).

Lemma 2.0.12

(i) Tnse% = e%sTns, T"e% = e%sT".(ii) If ( = (s then T 2

nse% = "Tnse%.

(iii) If ( != (s then T 2ns

e% = 0.

Proof. — Part (i) follows from Lemma 2.0.8. Lemma 2.0.6 (i) implies that it isenough to calculate T 2

nse%) = T 2

ns)%. Applying Lemma 2.0.6 (ii) twice we obtain

SuppT 2ns

)% + K. Hence it is enough to do the calculation in the space IndKI1 1. Since

K1 acts trivially on this space, it is enough to do the calculation in the space Ind!U 1.

Then the Lemma is a special case of [7] Theorem 4.4.

Lemma 2.0.13. — Let m ! 0 and let w = $ns then the following hold:(i) I1wI1wmI1 = I1wm+1I1,(ii) I1w"1I1wm+1 % I1wmI1 = I1wm,(iii) Twm = (Tw)m = (T"Tns)m.

Proof. — The first two parts can be checked by a direct calculation. For Part (iii) weobserve that

SuppTwTwm) + I1w Supp Twm) = I1wI1wmI1 = I1w

m+1I1,

where the last equality is Part (i). Part (ii) and Lemma 2.0.6 (ii) imply that

(TwTwm))(wm+1) = 1.

Since I1 acts trivially on ) and all the homomorphisms are G-equivariant, we mayapply Lemma 2.0.6 (ii) again to obtain

TwTwm) = Twm+1).

Lemma 2.0.6 (i) implies that TwTwm = Twm+1 . Induction and Lemma 2.0.8 gives usPart (iii).

Lemma 2.0.14

(i) Let n & N , then there exists h & H and integers a & {0, 1}, m ! 0 and b & Zsuch that

Tn = T a"(T"Tns)

mT b"Th

where T"1" = T""1 .


2.1. SUPERSINGULAR MODULES 11

(ii) The elements Tns , T", T""1 and e%, for every character ( : H # F!p , generate

H as an algebra.

Proof. — We note that Lemma 2.0.8 implies that T" is invertible with T"1" = T""1

and T 2" is central in H. Every n & N maybe written as n = $a($ns)m$bt, where

t & T . Lemma 2.0.8 and Lemma 2.0.13(iii) imply Part (i). Hence Tns , T", T""1 andTh, for h & H generate H as an algebra. Lemma 2.0.8 implies that The% = ((h"1)e%

and hence Lemma 2.0.10 (ii) implies that Th can be expressed as a linear combinationof idempotents e%. This gives us Part (ii).

Lemma 2.0.15

(i) The set {e%Tn) : n & N/T, ( : H # F!p } is an Fp-basis of (c-IndG

I1 1)I1 .(ii) The set {e%Tn : n & N/T, ( : H # F

!p } is an Fp-basis of H.

Proof. — Since e%Th = ((h"1)e% Lemma 2.0.6 (iii) implies that the set

{e%Tn) : n & N/T, ( : H "# F!p }

is a spanning set. Since the elements e% are orthogonal idempotents it is enough toshow that the set {e%Tn) : n & N/T } is linearly independent for a fixed charac-ter (. Lemma 2.0.6 (ii) implies that Supp e%Tn) = InI. Lemma 2.0.4 implies that ifnT != n&T , then e%Tn) and e%Tn#) have disjoint support and hence the set is linearlyindependent. Part (ii) follows from Part (i) and Lemma 2.0.6 (i).

2.1. Supersingular modules

All the irreducible modules of H have been determined by Vigneras in [18]. Theynaturally split up into two classes.

Proposition 2.1.1. — Let # be a smooth irreducible representation of G, which ad-mits a central character. Suppose that # is not supersingular, then #I1 is an irreducibleH-module.

Proof. — See [18] E.5.1.

The modules as above could be called non-supersingular, we are interested in allthe rest.

Definition 2.1.2. — Let ( : H # F!p be a character, let + = {(, (s} and let

, & F!p . We define a standard supersingular module M&

! to be a right H-module suchthat its underlying vector space is 2 dimensional

M&! = (v1, v2)Fp

and the action of H is determined by the following:



(i) If ( = (s then

v1e% = v1, v1Tns = "v1, v1T" = v2

and

v2e% = v2, v2Tns = 0, v2T" = ,v1.

(ii) If ( != (s then

v1e% = v1, v1Tns = 0, v1T" = v2

and

v2e%s = v2, v2Tns = 0, v2T" = ,v1.

To show that these relations define an action of H requires some work, this is donein [18].

Lemma 2.1.3. — The modules M&! are irreducible and

M&#

!# $= M&!

if and only if +& = + and ,& = ,.

Proof. — The definition immediately gives that M&! does not have a 1 dimensional

submodule, hence it is irreducible. If (& : H # F!p is a character, such that (& !& +

then

M&! e%# = 0.

Hence, + = +&. The element T 2" acts on M&

! by a scalar ,. Hence, , = ,&.

The following Proposition explains why M&! are called supersingular.

Proposition 2.1.4. — Let M be an irreducible H module, such that M !$= #I1 forany non-supersingular irreducible representation #, then

M $= M&!

for some + and ,.

Proof. — See [18] C.2 and E.5.1.

Corollary 2.1.5. — Let # be a smooth irreducible representation of G, admittinga central character. Suppose that #I1 contains a submodule isomorphic to M&

! forsome + and ,, then # is supersingular.

We will also need to consider the following extension of supersingular modules.


2.1. SUPERSINGULAR MODULES 13

Definition 2.1.6. — Let ( : H # F!p be a character, such that ( != (s, let + =

{(, (s} and let , & F!p . Let

H& = H/(T 2" " ,)H

then we define a right H-module L&! to be

L&! = e%H&/e%(T"Tns " TnsT")H&.

The definition seems to be asymmetric in ( and (s, however the multiplicationfrom the left by T" induces an isomorphism

e%H&/e%(T"Tns " TnsT")H& $= e%sH&/e%s(T"Tns " TnsT")H&,

since T" is a unit in H&.

Lemma 2.1.7. — The images of e%, e%T", e%Tns and e%TnsT" in L&! form an Fp-

basis of L&! .

Proof. — This follows from Lemma 2.0.15 (ii) and Lemma 2.0.12 (ii).

Lemma 2.1.8. — There exists a short exact sequence

0 "# M&! "# L&

! "# M&! "# 0

of H-modules.

Proof. — Let v1 be the image of e%Tns in L&! and let v2 be be image of e%TnsT" in

L&! . The subspace (v1, v2)Fp

is stable under the action of Tns , T" and e%# , for every

(& : H # F!p . Hence, by Lemma 2.0.14 (ii) the subspace is stable under the action

of H. From Lemma 2.0.12 (ii) and Definition 2.1.2 (ii) it follows that (v1, v2)Fp

$= M&! .

An easy check shows that L&!/M&

!$= M&

! .

Lemma 2.1.9. — Let (#,V) be a smooth representation of G and let - & F!p . Let µ'

be an unramified quasi-character:

µ' : F! "# F!p , x *"# -valF (x)

where valF is the valuation of F . Suppose that #I1 contains M&! , where + = {(, (s}

and let V be the underlying vector space of M&! in V. If we consider the representation

(# . µ' , det, V) of G, then the action of H on V is isomorphic to M&'"2

! .

Proof. — LetV = (v1, v2)Fp

as in Definition 2.1.2. Since µ' is unramified, Corollary 2.0.7 implies that the actionof Tns and the idempotents e% on V does not change. Lemma 2.0.14 (ii) implies thatit is enough to check how T" acts. Since det$ = "!F , twisting by µ' , det gives us

v1T" = $"1v1 = -"1v2 and v2T" = $"1v2 = -"1,v1.

Once we replace v1 by -v1 the isomorphism follows from Definition 2.1.2.



Since, by twisting by an unramified character we may vary , as we wish, we mightas well work with , = 1.

Definition 2.1.10. — Let + = {(, (s} then we define H-modules

M! = M1! and L! = L1

! .

2.2. Restriction to HK

Let HK = EndK(IndKI1 1). The natural isomorphism of K representations

IndKI1 1 $= {f & c-IndG

I1 1 : Supp f + K}

gives an embedding of algebras

HK %"# HomK(IndKI1 1, c-IndG

I1 1) $= HomG(c-IndGI1 1, c-IndG

I1 1) = H.

As an algebra HK is generated by Tns and e%, for all characters (.

Definition 2.2.1. — Let ( : H # F!p be a character. Let J0(() be a set, such that

J0(() = ! if ( != (s, and J0(() = {s}, if ( = (s. Let J be a subset of J0((), we defineM%,J to be a right HK-module, whose underlying vector space is one dimensional,M%,J = (v)Fp

and the action of HK is determined by the following:

ve% = v,

vTns = 0 if s & J or s !& J0((), vTns = "v, if s !& J and s & J0(().

Given ( and J as above, we will denote

J = J0(()\J.

Lemma 2.2.2. — Let ( : H # F!p be a character and let + = {(, (s}, then

M! |HK$= M%,J - M%s,J

as HK-modules, where J is a subset of J0((). Moreover, if ( != (s, then

L! |HK$= (IndK

I ( - IndKI (s)I1

as HK-modules.

Proof. — The first isomorphism follows directly from Definition 2.1.2. Since J0(()has at most two subsets, it doesn’t matter which subset we take. For the secondisomorphism we observe that the space (IndK

I ()I1 is two dimensional, with the basis{)%, Tns)%s}. Moreover, I acts on the basis vectors by characters ( and (s respec-tively. Now

)%Tns =%

u$I1/K1

un"1s )% = e%

! %

u$I1/K1

un"1s )

"= e%Tns) = Tnse%s) = Tns)%s


2.2. RESTRICTION TO HK 15

and

(Tns)%s)Tns =%

u$I1/K1

un"1s Tnse%s) = Tnse%s

! %

u$I1/K1

un"1s )

"= e%T 2

ns) = 0

and Lemma 2.1.7 allows us to define the obvious isomorphism on the basis.


CHAPTER 3

IRREDUCIBLE REPRESENTATIONS OF GL2(Fq)

3.1. Carter and Lusztig theory

In [7] Carter and Lusztig have constructed all irreducible Fp-representations of afinite group ", which has a ’split BN -pair of characteristic p’. Since GL2(Fq) is aspecial case of this, we will recall their results. Let " be a finite group with a BN -pair(", B, N, S). Let H = B % N , then H is normal in N , and S is the set of Coxetergenerators of W = N/H . We additionally require that B = HU , where U is a normalsubgroup of B, which is a p-group, and H is abelian of order prime to p. Moreover,we assume that H = %n$NBn.

Theorem 3.1.1 ([7]). — Let " be an irreducible representation of " then(i) the space of U invariants "U is one dimensional;(ii) suppose that the action of B on "U is given by a character ( : H # F

!p , via

B # B/U $= H and let J = {s & S : s " "U = "U} then the pair ((, J) determines "up to an isomorphism;

(iii) conversely, given a character ( : H # F!p , let J0(() = {s & S : (s = (} and

let J be a subset of J0(() then there exists an irreducible representation "%,J of " withthe pair ((, J) as above.

Proof. — This is [7] Corollary 7.5, written out in detail, see also [11] Theorem 3.9and [8] Theorem 4.3. and [6] §3.4.

Let H! = End!(Ind!U 1). We would like to rephrase Theorem 3.1.1 in terms of

H!-modules. For each s & S we may choose a representative ns & N . Moreover,according to [7] Lemma 2.2, we can choose ns in a nice way. The obvious equivalentof Definition 2.0.3 gives an endomorphism Tn & H! for each n & N . Definition 2.0.9for each character ( : H # F

!p gives an idempotent e% & H!.

Definition 3.1.2. — Let ( : H # F!p be a character, and let J be a subset of

J0(() we define M%,J to be a right H!-module, whose underlying vector space is one

18 CHAPTER 3. IRREDUCIBLE REPRESENTATIONS OF GL2(Fq)

dimensional, M%,J = (v)Fpand the action of H! is determined by the following:

ve% = v

and for every s & S we have

vTns =

()

*

0 if s & J ,"v if s & J0((), s !& J ,0 if s !& J0(().

Corollary 3.1.3. — The functor of U invariants

Rep! "# Mod -H!, " *"# "U

induces a bijection between the irreducible representations of " and the irreducibleright H!-modules. Moreover, if an irreducible representation "%,J corresponds to thepair ((, J), in the sense of Theorem 3.1.1 (iii), then

"U%,J

$= M%,J

as an H!-module.

Proof. — See, [6] Theorem 3.32.

Remark 3.1.4. — Ideally, we would like to have an analogue of the Corollary abovefor G or more generally for any group of F -points of a reductive group, split over F .

Carter and Lusztig, in [7] construct all the irreducible representations "%,J in avery elegant way. For each pair ((, J) they define a "-equivariant homomorphism

%Jw0

: Ind!B ( "# Ind!

B (w0

which depends on the geometry of the Coxeter group W , so that

"%,J$= Im %J

w0

where w0 is the unique element of maximal length in W .From now onwards we specialise to our situation, so that " = GL2(Fq), B is

the subgroup of upper-triangular matrices, U is the subgroup of unipotent upper-triangular matrices, H is the diagonal matrices, N is the normaliser of H in ", thatis the monomial matrices and W = N/H is isomorphic to the symmetric group ontwo letters, W = {1, s}. Let

ns =!

0 "11 0

"

be a fixed representative of s in N . In particular, s is the element of the maximallength in W and also the single Coxeter generator, so that S = {s}. Hence, if( : H # F

!p , then either J0(() = ! or J0(() = S. Since

K/K1$= ", I/K1

$= B, I1/K1$= U

to ease the notation, we will often identify the spaces

{f : " "# Fp : f(ug) = f(g), ' g & ", 'u & U}


3.1. CARTER AND LUSZTIG THEORY 19

and{f & c-IndG

I1 1 : Supp f + K}in the natural way. In particular, we will use the same notation for the elements ofHK and H! and we note that the Definitions 2.2.1 and 3.1.2 coincide.

Proposition 3.1.5. — For each character ( : H # F!p , such that ( = (s, let

"%,S = Im+(1 + Tns) : Ind!

B ( # Ind!B (

,

and let"%,! = Im

+Tns : Ind!

B ( # Ind!B (

,

then the representations "%,S and "%,! are irreducible. Moreover,

"U%,S = ((1 + Tns))%)Fp

$= M%,S and "U%,! = (Tns)%)Fp

$= M%,!

as H!-modules. For each character ( : H # F!p , such that ( != (s, let

"%,! = Im+Tns : Ind!

B ( # Ind!B (s

,

then the representation "%,! is irreducible. Moreover,

"U%,! = (Tns)%)Fp

$= M%,!

as an H!-module. Further, these representations are pairwise non-isomorphic, andevery irreducible representation of " is isomorphic to "%,J , for some character ( anda subset J of J0(().

Proof. — This is a special case of [7] Theorem 7.1 and Corollary 7.5. The isomor-phisms of H!-modules are given by the Corollary 3.1.3.

Remark 3.1.6. — Although we do not use this, we note that Frobenius reciprocitygives us

c-IndGK "%,! $= Tns(c-IndG

I () # c-IndGI (s

and if ( = (s then

c-IndGK "%,S

$= (1 + Tns)(c-IndGI () # c-IndG

I (.

Using this, one can relate the central elements of Vigneras in [18] to the ‘standard ’endomorphisms T# of Barthel and Livne in [1].

Lemma 3.1.7. — Let ( : H # F!p be a character, such that ( = (s. Then the homo-

morphisms e%(1 + Tns)e% and "e%Tnse% are orthogonal idempotents. In particular,

Ind!B ( $= "%,! - "%,S .

Moreover, let (& : F!q # F

!p be a character such that ( = (& , det, then

"%,S$= (& , det and "%,! $= St.(& , det

where St is the Steinberg representation.



Proof. — Since ( = (s we have

e%Tns = Tnse% and e%T 2ns

= "e%Tns .

So the elements above are orthogonal idempotents as claimed. By Proposition 3.1.5,the summands they split o# are "%,S and "%,!.

Since ( = (s, the character ( must factor through the determinant. So ( extendsto a character of " and hence

Ind!B ( $= Ind!

B 1. (& , det .

So we may assume that ( is the trivial character. The Bruhat decomposition says that" = BsB / B and hence by Theorem 3.1.1 (ii) "1,S = 1, the trivial representationsof G. This implies that "1,! is the Steinberg representation.

Corollary 3.1.8. — Let ( : H # F!p be a character, such that ( = (s. Let " be

any representation of ", such that for some v & "U we have

(v)Fp

$= M%,J

as an H!-module. Then("v)Fp

$= "%,J

as a "-representation.

Proof. — Since v is fixed by U there exists a homomorphism * & Hom!(Ind!U 1, ")

such that *()) = v. The isomorphism of H!-modules implies that

v = ve% = *(e%)) = *()%).

Hence, H acts on v by a character ( and

*(Ind!U 1) = *(e%(Ind!

U 1)) = *(Ind!B ().

If J = ! then*((1 + Tns))%) = v(1 + Tns)e% = 0.

Hence, "%,S is contained in the kernel of *. By Lemma 3.1.7

Im * $= "%,!.

Since, the image is irreducible and contains v we get the result. The proof for J = Sis analogous.

The Corollary has a nice application, which complements [18] E.7.1.

Corollary 3.1.9. — Let # be a smooth representation of G and suppose that thereexists a non-zero vector v & #I1 such that

ve1 = v, vTns = 0, vT" = v

then G acts trivially on v.


3.2. ALTERNATIVE DESCRIPTION OF IRREDUCIBLE REPRESENTATIONS 21

Proof. — As an HK module(v)Fp

$= M1,S .

By Corollary 3.1.8 K acts trivially on v. On the other hand

v = vT" = $"1v.

Iwahori decomposition implies that $ and K generate G as a group. Hence G actstrivially on v.

Remark 3.1.10. — There is a version of this twisted by a character. This examplewill lead us to better things. See Remark 5.5.6.

Lemma 3.1.11. — Let ( : H # F!p be a character, let J be a subset of J0((), and

let J = J0(()\J . The sequence of H!-modules

0 "# M%,J "# (Ind!B (s)U "# M%s,J "# 0

is exact. Moreover, it splits if and only if ( = (s.

Proof. — The space (Ind!B (s)U is two dimensional, with the basis {Tns)%, )%s}.

If ( = (s then e%(1 + Tns)e% and "e%Tnse% are orthogonal idempotents, whichsplit the sequence.

If ( != (s then J0(() = J = ! and for every ,, µ & Fp we have

(,Tns)% + µ)%s)e% = ,Tns)%, (,Tns)% + µ)%s)e%s = µ)%s

and(,Tns)% + µ)%s)Tns = µTns)%.

Hence M%,! is the only proper submodule, so the sequence cannot split.

3.2. Alternative description of irreducible representations

Let Vd,F be an F vector space of homogeneous polynomials in two variables Xand Y of the degree d. The group K acts on Vd,F via

!a bc d

"(Xd"iY i) = (aX + cY )d"i(bX + dY )i.

For 0 # i # d, let

mi =!

di

"Xd"iY i

where+

di

,denotes the binomial coe!cient. Vectors mi, for 0 # i # d, form a basis of

Vd,F . Let Vd,oF be the oF -lattice in Vd,F spanned by the mi, for 0 # i # d. An easycheck shows that Vd,oF is K invariant. Let

Vd,Fq = Vd,oF .oF oF /pF .



The vectors mi . 1, for 0 # i # d, form an Fq-basis of Vd,Fq . The subgroup K1

acts trivially on Vd,Fq , so we consider Vd,Fq as a representation of ". Let Fr be theautomorphism of ", given by

Fr :!

a bc d

"*"#

!ap bp

cp dp

".

Let " be a representation of ". We will denote by "Fr the representation of " givenby

"Fr(g) = "(Fr(g)).

Theorem 3.2.1. — Let " = GL2(Fq) and suppose that q = pn. The isomorphismclasses of irreducible Fp-representations of " are parameterised by pairs (a, r), where

– a is an integer 1 # a # q " 1 and– r is an ordered n-tuple r = (r0, r1, . . . , rn"1), where 0 # ri # p " 1, for every i.

Moreover, the irreducible representations of " can be realized over Fq and the irre-ducible representation corresponding to (a, r) is given by

Vr,Fq . (det)a $= Vr0,Fq . V Frr1,Fq

. · · ·. V Fri

ri,Fq. · · ·. V Frn"1

rn"1,Fq. (det)a.

Proof. — This is shown in [3], see also [1] Proposition 1 and [18] Ap. 6. We remarkthat since

+ri

,is a unit in Fq if r # p " 1, our spaces really coincide with the ones

considered in [1].

We fix some embedding $ : Fq %# Fp and we will assume that every character( : H # F

!p factors through $. Once we have done that, we will omit $ from our

notation. We will denoteVr,Fp

= Vr,Fq .Fq Fp.

We need a dictionary between the two descriptions.

Proposition 3.2.2. — Let ( : H # F!p and let a be the unique integer, such that

1 # a # q " 1 and(++

1 00 &

,,= ,a ', & F!

q

and let r be the unique integer, such that 1 # r # q " 1 and

(++

& 00 &"1

,,= ,r ', & F!

q .

Suppose that r != q " 1, and let r = (r0, . . . , rn"1) be the digits of a p-adic expansionof r

r = r0 + r1p + · · · + rn"1pn"1

then ( != (s and "%,! corresponds to the pair (a, r). More precisely

"%,! $= Vr0,Fp. · · ·. V Frn"1

rn"1,Fp. (det)a.

Suppose that r = q " 1, then ( = (s,

"%,! $= Vp"1,Fp. · · ·. V Frn"1

p"1,Fp. (det)a $= St.(det)a


3.2. ALTERNATIVE DESCRIPTION OF IRREDUCIBLE REPRESENTATIONS 23

and"%,S

$= V0,Fp. · · ·. V Frn"1

0,Fp. (det)a $= (det)a

where St denotes the Steinberg representation.

Proof. — Every character ( : H # F!p is of the form

( :!

, 00 µ

"*"# ,cµd

for some integers c and d. Moreover, ( = (s if and only if

c " d 0 0 (mod q " 1).

The integers a and r are uniquely determined by the congruences

d 0 a (mod q " 1) and c " d 0 r (mod q " 1).

By Theorem 3.1.1 if " is an irreducible representation of ", then dim "U = 1, andby Corollary 3.1.3 the irreducible representations of " correspond to the irreduciblemodules of the Hecke algebra H!. Since we have two complete lists of irreduciblerepresentations, it is enough to match up the corresponding irreducible modules. Werecall that

"U%,J

$= M%,J

as H!-modules.We observe that the action of U on Vd,Fp

fixes the vector m0 . 1. Moreover,!

, 00 µ

"m0 . 1 = ,dm0 . 1.

Let (a, r) be any pair parameterising an irreducible representation of " and let

r = r0 + r1p + · · · + rn"1pn"1.

By picking such (m0 . 1)ri in every component of the tensor product we obtain anon-zero vector

(m0 . 1)r = (m0 . 1)r0 . · · ·. (m0 . 1)rn"1

fixed by U . The vector (m0 . 1)r spans the space of U invariants, since it is onedimensional. Moreover, since the action on the components of the tensor product istwisted by Fr we obtain

!, 00 µ

"(m0 . 1)r = (,µ)a,r(m0 . 1)r.

Suppose that we start with an arbitrary character ( : H # F!p and obtain the

integers a and r as in the statement of the proposition.If r != q"1, then by above ( != (s. Let r be the n-tuple corresponding to r. Since,

( != (s, the module M%,! is the only irreducible module of H!, which is not killed by



the idempotent e%. Let (m0 . 1)r be the vector constructed above. Since, H acts on(m0 . 1)r via the character (, we obtain

M%,! $= (Vr0,Fp. · · ·. V Frn"1

rn"1,Fp. (det)a)U

as H!-modules and that implies the isomorphism between representations.If r = q " 1, then ( = (s, and the only H!-modules, which are not killed by e%,

are M%,S and M%,!. We observe that V0,Fpis just the trivial representation. Let 0 =

(0, . . . , 0), then the representation corresponding to the pair (a,0) is just 1. (det)a,which is isomorphic to "%,S , by Proposition 3.1.7. The only case left is r = p " 1 =(p " 1, . . . , p " 1), hence

M%,! $= (Vp"1,Fp. · · ·. V Frn"1

p"1,Fp. (det)a)U

as H!-modules, since the module M%,S is already taken. This implies that

"%,! $= Vp"1,Fp. (det)a $= St.(det)a

where the last isomorphism follows from Proposition 3.1.7.

Corollary 3.2.3. — Suppose that q = pn and the representation "%,J correspondsto the pair (a, r). Let r = r0 + r1p + · · · + rn"1pn"1 and let J = J0(()\J , whereJ0(() = {s & S : (s = (}. Then

"%s,J$= Vp"1"r0,Fp

. · · ·. V Frn"1

p"1"rn"1,Fp. (det)a+r.

Proof. — If r = 0 or r = q"1, then r is of a special form and the isomorphism followsfrom Proposition 3.2.2.

If r != 0 and r != q " 1, we observe that

(s++

1 00 &

,,= (

++& 00 &"1

,,(++

1 00 &

,,= ,a+r ', & F!

q

and(s

++& 00 &"1

,,= (

++&"1 00 &

,,= ,"r ', & F!

q .

The claim follows from Proposition 3.2.2.


CHAPTER 4

PRINCIPAL INDECOMPOSABLE REPRESENTATIONS

We will recall some facts from the modular representation theory of finite groups.Let " be any finite group. We denote by Rep! the category of Fp-representations of" and by Irr! the set of isomorphism classes of irreducible representations in Rep!.We note that Rep! is equivalent to the module category of the ring Fp["].

Proposition 4.0.4. — A representation inj is an injective object in Rep! if and onlyif it is a projective object in Rep!.

The isomorphism classes of indecomposable injective (and hence projective) objectsin Rep! are parameterised by Irr!.

More precisely, if inj is indecomposable and injective, then the maximal semi-simplesubmodule soc(inj) and the maximal semi-simple quotient inj / rad(inj) are both irre-ducible. Moreover,

soc(inj) $= inj / rad(inj).

Conversely, given " & Irr!, there exists a unique up to isomorphism indecomposable,injective object inj " in Rep!, such that

" $= soc(inj ").

Proof. — See [15], Exercises 14.1 and 14.6.

We will call indecomposable representations of ", which are injective objects inRep!, principal indecomposable representations.

Remark 4.0.5. — We note that a monomorphism " %# inj " is an injective envelopeof " in Rep!.

Corollary 4.0.6. — We have the following decomposition:

Fp["] $=-

$$Irr!

(dim ") inj ".

26 CHAPTER 4. PRINCIPAL INDECOMPOSABLE REPRESENTATIONS

Proof. — Since Fp["] is an injective and projective object it must decompose into adirect sum of indecomposable injective objects. Since

dimHom!(",Fp["]) = dimHom{1}(",1) = dim "

the representation inj " occurs in the decomposition with the multiplicity dim ".

Proposition 4.0.7. — Let U be a p-Sylow subgroup of ". Then a representation "is an injective object in Rep! if and only if "|U is an injective object in RepU .

Proof. — This follows easily from [15], §14.4, Lemma 20.

Proposition 4.0.8. — Suppose that U is a p-group, then the only irreducible repre-sentation is 1 and hence the only principal indecomposable representation is Fp[U ].

Proof. — The first part is [15], §8 , Proposition 26, the last part follows from Corol-lary 4.0.6.

Corollary 4.0.9. — Let inj be an injective object in Rep! and let U be a p-Sylowsubgroup of ", then

dim inj = dim injU |U |.

Proof. — The restriction inj |U is an injective object in RepU . By the above Propo-sition

inj |U$= mFp[U ].

The multiplicity m is given by: m = dim HomU (1, inj) = dim injU .

In the rest of the section " = GL2(Fq) and U is the subgroup of unipotent uppertriangular matrices. Given " & Irr! we are going to compute (inj ")U as an H!-module. Once we know the modules we are going to show that if we consider inj "%,J

and inj "%s,J as representations of K, then the action of HK on

(inj "%,J - inj "%s,J)I1

extends to the action of H, so that if ( = (s then it is isomorphic to a direct sum ofsupersingular modules and if ( != (s then it is isomorphic to a direct sum of L! andsupersingular modules. See Propositions 4.2.37 and 4.2.38 for the precise statement.This calculation, becomes of importance in Section 6.4. Although, the general caseincludes the case q = p, if q = p we give a di#erent, easier way of doing this. Whenq = p, the main result is Proposition 4.1.9.


4.1. THE CASE q = p 27

4.1. The case q = p

We start o# with no assumption on q.

Lemma 4.1.1. — Suppose that ( != (s, then there exists an exact sequence

0 "# Ind!B (s *""# inj "%,!

of "-representations.

Proof. — Since inj "%,! is an injective module, there exists * such that the diagram

0 !! "%,! !!

""

Ind!B (s

*##

inj "%,!

commutes. If Ker * != 0, then (Ker*)U is a non-zero proper submodule of (Ind!B (s)U

not containing M%,!. By Lemma 3.1.11 this cannot happen.

Corollary 4.1.2. — Suppose that ( != (s then

dim inj "%,! ! 2q.

Proof. — Corollary 4.0.9 implies that

dim inj "%,! = dim(inj "%,!)U |U |.

The order of U is q and since by Lemma 4.1.1 Ind!B (s is a subspace of inj "%,!, we

obtaindim(inj "%,!)U ! 2.

Lemma 4.1.3. — Suppose that q = p and ( != (s then the sequence of " representa-tions

0 "# "%,! "# Ind!B (s "# "%s,! "# 0

is exact.

Remark 4.1.4. — This fails if q != p.

Proof. — The argument below is taken from [18] Ap. 6. We know that

"%s,! $= Tns(Ind!B (s)

and "%,! is isomorphic to the subspace of Ind!B (s generated by Tns)%. Since, T 2

ns)% =

0 we always have"%,! # KerTns .

If q = p, then by Proposition 3.2.2 and Corollary 3.2.3 there exists an integer r suchthat

dim "%,! + dim "%s,! = (r + 1) + (p " 1 " r + 1) = p + 1 = dim Ind!B (s.

Hence the sequence is exact.



Corollary 4.1.5. — Suppose that q = p and let ( : H # F!p be a character, such

that ( != (s. Let " be any representation of ", such that for some v & "U

(v)Fp

$= M%,!

as an H!-module. Then("v)Fp

$= "%,!


Remark 4.1.6. — This fails if p != q, by Remark 4.1.4, it is enough to look atInd!

B (/"%s,!.

Proof. — Since v is fixed by U , there exists a homomorphism * & Hom!(Ind!U 1, ")

such that *()) = v. The isomorphism of H!-modules implies that

v = ve% = *(e%)) = *()%).

Hence H acts on v by a character ( and

*(Ind!U 1) = *(e%(Ind!

U 1)) = *(Ind!B ().

Now*(Tns)%s) = vTnse%s = 0.

Hence, "%s,! is contained in the kernel of *. By Lemma 4.1.3

Im * $= "%,!.

Since, the image is irreducible and contains v we get the result.

Lemma 4.1.7. — Suppose that q = p. If ( = (s then

dim inj "%,J = p.

If ( != (s thendim inj "%,! = 2p.

Proof. — Corollary 4.0.6 implies that

dimFp["] =%

$$Irr!

(dim ")(dim inj ")

=%

%,%=%s

(dim "%,!)(dim inj "%,!) + (dim "%,S)(dim inj "%,S)

+12

%

%,%'=%s

(dim "%,!)(dim inj "%,!) + (dim "%s,!)(dim inj "%s,!).

If ( = (s then Corollary 4.0.9 implies that

dim inj "%,J ! p.

If ( != (s then Corollary 4.1.2 implies that

dim inj "%,! ! 2p.


4.1. THE CASE q = p 29

Lemma 4.1.3 and Lemma 3.1.7 imply that

dim "%,J + dim "%s,J = p + 1.

We put these inequalities together and we obtain

dimFp["] !%

%

(p + 1)p = dimFp["]

So all the inequalities must be equalities and we obtain the lemma.

Corollary 4.1.8. — Suppose that q = p. If ( = (s then

("(inj "%,J )U )Fp

$= "%,J .

In particular,(inj "%,J )U $= M%,J

as an H!-module.If ( != (s then

("(inj "%,!)U )Fp$= Ind!

B (s.

In particular,(inj "%,J)U $= (Ind!

B (s)U

as an H!-module.

Proof. — If ( = (s then we have an exact sequence

0 "# "%,J "# inj "%,J

of "-representations. Since, by Lemma 4.1.7

dim "U%,J = dim(inj "%,J )U

we obtain the Corollary. Similarly, if ( != (s then by Lemma 4.1.1 there exists anexact sequence

0 "# Ind!B (s "# inj "%,!

of "-representations. Since, by Lemma 4.1.7

dim(Ind!B (s)U = dim(inj "%,!)U

we obtain the Corollary.

Proposition 4.1.9. — Suppose that q = p, let ( : H # F!p be a character and

let + = {(, (s}. We consider representations inj "%,J and inj "%,J as representationsof K, via

K "# K/K1$= ".

If ( = (s then the action of HK on (inj "%,! - inj "%,S)I1 extends to the action of Hso that

(inj "%,! - inj "%,S)I1 $= M! .



If ( != (s then the action of HK on (inj "%,! - inj "%s,!)I1 extends to the action of Hso that

(inj "%,! - inj "%s,!)I1 $= L! .

Proof. — Suppose that ( = (s then Corollary 4.1.8 says that

(inj "%,! - inj "%,S)I1 $= (Tns)%)Fp- ((1 + Tns))%)Fp

$= M%,! - M%,S$= M! |HK

as HK-modules, where the last isomorphism follows from Lemma 2.2.2. It is enoughto define the action of T". If we let

(Tns)%)T" = (1 + Tns))% and ((1 + Tns))%)T" = Tns)%

then this gives us the required action. Suppose that ( != (s, then Corollary 4.1.8 andLemma 2.2.2 imply that

(inj "%,! - inj "%s,!)I1 $= (IndKI (s - IndK

I ()I1 $= L! |HK

as HK-modules. The space (IndKI (s)I1 has basis {Tns)%, )%s} and the space

(IndKI ()I1 has basis {Tns)%s , )%}. It is enough to define the action of T" on the

basis. If we set)%T" = )%s , )%sT" = )%

and(Tns)%)T" = Tns)%s , (Tns)%s)T" = Tns)%

then this gives us the required action.

4.2. The general case

Our counting argument breaks down if p != q. The strategy is to restrict to SL2(Fq),where the principal indecomposable representations have been worked out by Jeyaku-mar in [10]. Let

"& = SL2(Fq), B& = B % "&, H & = H % "&.

We note that U is a subgroup of "& and ns & "&.

4.2.1. Modular representations of SL2(Fq). — The irreducible Fp-representa-tions of SL2(Fq) were determined by Brauer and Nesbitt.

Theorem 4.2.1 ([3]). — Suppose that q = pn. The isomorphism classes of irre-ducible Fp-representations of "& are parameterised by n-tuples r = (r0, . . . , rn"1),where 0 # ri # p " 1, for every i. Moreover, every irreducible representation can berealized over Fq and the representation corresponding to an n-tuple r is given by

Vr,Fq$= Vr0,Fq . V Fr

r1,Fq. · · ·. V Fri

ri,Fq. · · ·. V Frn"1

rn"1,Fq

where Vri,Fq are the spaces of Section 3.2.


4.2. THE GENERAL CASE 31

Corollary 4.2.2. — Let " be an irreducible representation of ", then " |!# is irre-ducible. Moreover, given an irreducible representation "& of "& there exist, preciselyq " 1 isomorphism classes of irreducible representations of ", given by " . (det)a,where 0 # a < q " 1, such that

(" . (det)a) |!#$= "&.

Proof. — This is immediate from Theorem 4.2.1 and Theorem 3.2.1.

Remark 4.2.3. — By counting dimensions, we may show that

(inj(Vr,Fp. (det)a)) |!#$= inj Vr,Fp

as "&-representations. However, we will obtain this directly later on.

We recall the construction of the indecomposable principal representations forSL2(Fq) as it is done in [10]. The idea is to go from the Lie algebra to the uni-versal enveloping algebra and then to the group.

Let g be the Lie algebra of SL2(C). It has a C-basis

e =!

0 10 0

", h =

!1 00 "1

", f =

!0 01 0

".

Let U be the universal enveloping algebra of g. Let UZ be a subring of U generatedby the elements

ek

k!,

fk

k!, ' k & Z+

over Z. The ring UZ has a Z-basis, which is also a C-basis for U . Let d be a non-negative integer and let Vd be the irreducible module of g of highest weight d. Thespace Vd has a C-basis of weight vectors mi, for 0 # i # d, and the action of g is givenby

em0 = 0, emi = (d " i + 1)mi"1, 1 # i # d,

fmd = 0, fmi = (i + 1)mi+1, 0 # i # d " 1,

hmi = (d " 2i)mi, 0 # i # d.

Let Vd,Z be a Z-lattice in Vd spanned by mi, for 0 # i # d. We adopt the conventionthat mi = 0 if i < 0 or i > d. Since,

ek

k!mi =

!d " i + k

d " i

"mi"k

andfk

k!mi =

!i + k

i

"mi+k

for all k & Z+, the lattice Vd,Z is a UZ-module. Let

#Vd,Fq = Vd,Z .Z Fq.



For every , & Fq we define x(,), y(,) & End(#Vd,Fq), by

x(,)(v . 1) =%

k!0

,k.ek

k!v . 1

/

and

y(,)(v . 1) =%

k!0

,k.fk

k!v . 1

/.

Since e and f act nilpotently on Vd this sum is well defined. There exists a uniquehomomorphism

SL2(Fq) "# End(#Vd,Fq)such that !

1 ,0 1

"*"# x(,) and

!1 0, 1

"*"# y(,).

This gives us a representation of "&. To ease the notation, we denote

mi,Fq = mi . 1.

We will refer to {mi,Fq : 0 # i # d} as the standard basis of #Vd,Fq . The action of "&

is determined by!

1 ,0 1

"mi,Fq =

i%

k=0

!d " kd " i

",i"kmk,Fq ,

!1 0, 1

"mi,Fq =

d%

k=i

!ki

",k"imk,Fq .

This gives !, 00 ,"1

"mi,Fq = ,d"2imi,Fq .

At first we resolve the ambiguities in our notation.

Lemma 4.2.4. — Let Vd,Fq be a representation of " constructed in Section 3.2. Then

Vd,Fq |!# $= #Vd,Fq .

Proof. — The isomorphism is given by

mi . 1 *"# mi,Fq .

An easy check shows that the isomorphism respects the action of matrices+

1 &0 1

,and+

1 0& 1

,, for all , & Fq. Since, these matrices generate "& we are done.

The Lemma above is the reason, why we wanted to work over Fq. We drop thetilde from our notation and go to Fp.

For each r, such that 0 # r < p" 1, Jeyakumar finds a "&-invariant subspace Rr ofthe representation Vp"1"r,Fp

. Vp"1,Fp, such that dim Rr = 2p. Let Rp"1 = Vp"1,Fp

,then dimRp"1 = p. The main result of [10] can be stated as follows.



Theorem 4.2.5 ([10]). — Suppose that q = pn. Let r = (r0, . . . , rn"1) be an n-tuple,such that 0 # ri # p " 1, for every i. Let

Rr = Rr0 . RFrr1

. · · ·. RFrn"1

rn"1.

If r != 0, thenRr

$= inj Vr,Fp.

AndR0

$= inj V0,Fp- injVp"1,Fp

where p " 1 = (p " 1, . . . , p " 1) and 0 = (0, . . . , 0).

Remark 4.2.6. — Our indices di#er slightly from [10].

4.2.2. Going from SL2(Fq) to GL2(Fq). — We will recall how the subspaces Rr

are constructed and show that they are in fact "-invariant. That this should be thecase is indicated by Remark 4.2.3. The twisted tensor product will give us principalindecomposable representations of ". Since the spaces Rr have a rather concretedescription, this will enable us to work out the corresponding H!-modules.

Lemma 4.2.7. — Let V be a representation of " and let W be a "&-invariant subspaceof V . If W is invariant under the action of H, then W is "-invariant.

Proof. — Let v & W and g & ". We may write g = g&g1, for some g& & "& and g1 & H .Then

gv = g&(g1v) & W.

Hence W is "-invariant.

Let r be an integer such that 0 # r # p" 1. Let {vi}, for 0 # i # p" 1 " r be thestandard basis of Vp"1"r,Fp

and let {wj}, for 0 # j # p " 1 be the standard basis ofVp"1,Fp

.

Definition 4.2.8. — For 0 # i # 2p " r " 2, we define vectors Ei in Vp"1"r,Fp.

Vp"1,Fp, by

Ei =%

k+l=i

vk . wl.

It is convenient to extend the indexing set to Z by setting Ei = 0, if i < 0 ori > 2p " 2p " r.

Lemma 4.2.9. — The sequence of "-representations

0 "# V2p"r"2,Fp"# Vp"1"r,Fp

. Vp"1,Fp

mi,Fp*"# Ei

is exact.



Proof. — If r = p " 1 then this is true trivially. If r != p " 1 then the map is "&-equivariant by [10] Lemma 4.2. So by Lemma 4.2.7 it is enough to show that it isH-equivariant. Since !

, 00 µ

"mi,Fp

= ,2p"r"2"iµimi,Fp

and !, 00 µ

"Ei = ,2p"r"2"iµiEi

we are done.

Definition 4.2.10 ([10]). — Let r be an integer, such that 0 # r < p " 1. For0 # i # p " r " 1, let ai be integers defined by the following relation:

a0 = 0 and a1 = (p " r " 2)!

and

ai+1 = ai +("1)i(r + 1) . . . (r + i)

(p " r " 2) . . . (p " r " i " 1)(a1 " a0).

Let Z be a vector in Vp"1"r,Fp. Vp"1,Fp

given by

Z = a0(v0 . wp"r"1) + a1(v1 . wp"r"2) + · · · + ap"r"1(vp"r"1 . w0),

and let Rr be a subspace of Vp"1"r,Fp. Vp"1,Fp

given by

Rr =0E0, . . . , E2p"r"2, Z,

f

1!Z, . . . ,

f r

r!Z

1

Fp

.

Moreover, for r = p " 1 we define

Rp"1 = Vp"1,Fp.

Proposition 4.2.11. — Let r be an integer, such that 0 # r # p " 1, then Rr is a"-invariant subspace of Vp"r"1,Fp

. Vp"1,Fp. Moreover, if r != p " 1, then

dim Rr = 2p

and if r = p " 1, thendim Rp"1 = p.

Proof. — If r = p"1 then there is nothing to prove, since Rp"1 = Vp"1,Fp. If r != p"1

then by [10] Theorem 4.7 Rr is "&-invariant and dimRr = 2p. So by Lemma 4.2.7it is enough to show that Rr is H-invariant. For v & Vp"r"1,Fp

and w & Vp"1,Fpwe

havef(v . w) = fv . w + v . fw.

Hence, for 0 # k # r we have

fk

k!Z &

2vl+i . wp"r"1"l+j | i + j = k, 0 # l # p " r " 1

3Fp



with the usual ’vanishing when not defined’ convention. Since!

, 00 µ

"vl+i . wp"r"1"l+j = ,p"r"1"l"iµl+i,r+l"jµp"r"1"l+jvl+i . wp"r"1"l+j

= ,p"k"1µp"r"1+kvl+i . wp"r"1"l+j

the group H acts on each fk

k! Z, for 0 # k # r by a character. We combine this withLemma 4.2.9 and obtain that Rr is H invariant.

Lemma 4.2.12. — We havefk

k!Ep"r"1 = 0

if and only if k ! r + 1. For k ! 1 we haveek

k!Ep"r"1 = 0.

In particular, U fixes Ep"r"1 and the action of H is given by!

, 00 µ

"Ep"r"1 = ,r(,µ)p"r"1Ep"r"1.

Proof. — If r = p " 1, then this is trivial. If r != p " 1 then for k ! 0 we have

fk

k!Ep"r"1 =

!p " r " 1 + k

p " r " 1

"Ep"r"1+k.

We observe that Ep"r"1+k vanishes trivially, if k ! p. If r + 1 # k # p " 1, then wewrite k = r + 1 + j, where 0 # j # p " r " 2. The binomial coe!cient becomes

!j + p

p " r " 1

".

Since 0 # r < p " 1, we have 1 # p " r " 1 # p " 1, and since 0 # j < p " r " 1, pdivides the binomial coe!cient. Hence

fk

k!Ep"r"1 = 0

for k ! r + 1. If 0 # k # r, then p " r " 1 # p " r " 1 + k # p " 1 and the binomialcoe!cient does not vanish. Hence

fk

k!Ep"r"1 != 0

for 0 # k # r. Let k ! 0, thenek

k!Ep"r"1 =

!p " 1 + k

p " 1

"Ep"r"1"k.

We observe that Ep"r"1"k vanishes trivially, if k > p " r " 1. Suppose that 1 # k #p" r"1, then we may write k = j"1, where 0 # j # p" r"2 < p"1. The binomialcoe!cient becomes !

j + pp " 1

".



Since j < p " 1, p divides the binomial coe!cient, and hence

ek

k!Ep"r"1 = 0

for all k ! 1. Since the action of U is given in terms of ek/k! this implies that U fixesEp"r"1. An easy verification gives us the action of H .

Proposition 4.2.13. — Let Wr be a subspace of Rr given by

Wr = (Ep"r"1, . . . , Ep"1)Fp.

Then Wr is "-invariant. Moreover,

WUr = (Ep"r"1)Fp

andWr = ("Ep"r"1)Fp

$= Vr,Fp. (det)p"r"1.

Proof. — If r = p " 1 then Wp"1$= Vp"1,Fp

and we are done. Otherwise, since Wr

has a basis of eigenvectors for the action of H , it is enough to show that Wr is "&-invariant. Since the action of "& is given in terms of the action of UZ it is enough toshow that Wr is invariant under the action of UZ. Lemma 4.2.12 implies that Wr hasa basis fk

k! Ep"r"1, for 0 # k # r. We observe that Lemma 4.2.12 also implies that

f l

l!(fk

k!Ep"r"1) =

!k + l

k

"fk+l

(k + l)!Ep"r"1 & Wr

for 0 # k # r and l ! 0. Suppose that 0 # k # r and l ! k + 1 then

el

l!(fk

k!Ep"r"1) = 0.

This follows from the multiplication in UZ, see [9] §26.2, and Lemma 4.2.12. If0 # l # k # r, then

el

l!(fk

k!Ep"r"1) =

!p " r " 1 + k

p " r " 1

"el

l!Ep"r"1+k

=!

p " r " 1 + kp " r " 1

" !p " 1 " k + l

p " 1 " k

"Ep"r"1+k"l & Wr .

Hence Wr is invariant under the action of UZ and hence under the action of ".We know from Lemma 4.2.12 that Ep"r"1 is fixed by U . The action of H splits

WUr into a direct sum of one dimensional subspaces. Suppose that dim WU

r ! 2. SinceH acts on each vector Ep"r"1+k by a distinct character for 0 # k # r, we must haveEp"r"1+j & WU

r , for some 1 # j # r. This implies that

eEp"r"1+j = (p " j)Ep"r"2+j = 0.

Hence p must divide j and this is impossible. Hence, dim WUr = 1.



Since Wr is "-invariant, we have

("Ep"r"1)Fp# Wr .

We may choose r + 1 distinct elements ,i in Fq. Then!

1 0,i 1

"Ep"r"1 =

r%

k=0

,kifk

k!Ep"r"1.

Let A be an (r + 1) 1 (r + 1) matrix, given by Aki = ,ki , for 0 # i, k # r, with

the convention that 00 = 1. Then detA is the Vandermonde determinant, which isnon-zero, since all the ,i are distinct. Hence, A is invertible and

fk

k!Ep"r"1 & ("Ep"r"1)Fp

for all 0 # k # r. Hence, Wr = ("Ep"r"1)Fp.

Since dim WUr = 1 and Wr = ("WU

r )Fp, the representation Wr is irreducible. To

decide, which one it is, we may proceed as in the proof of Proposition 3.2.2. Sincer < p " 1, the action of B on WU

r implies that Wr$= Vr,Fp

. (det)p"r"1.

Lemma 4.2.14. — The vector E0 is fixed by the action of U . Moreover, H acts onE0 by !

, 00 µ

"E0 = ,r(,µ)p"r"1(,µ"1)p"r"1E0.

Proof. — Since E0 = v0 . w0 this is immediate.

Definition 4.2.15. — Suppose that q = pn and let r = (r0, . . . , rn"1) be the n-tuplesuch that 0 # ri # p " 1, then we define a representation Rr of ", given by

Rr = Rr0 . RFrr1

. · · ·. RFrn"1

rn"1

where Rri are "-representations of Definition 4.2.10.

Definition 4.2.16. — Suppose that q = pn and let r = (r0, . . . , rn"1) be an n-tuple,such that 0 # ri # p " 1, for every i. Let ! = (.0, . . . , .n"1) be an n-tuple, such that.i & {0, 1} for every i. We define a vector

b! = E(1"(0)(p"1"r0) . · · ·. E(1"(n"1)(p"1"rn"1)

in Rr, where E(1"(i)(p"1"ri) is a vector in Rri , for each 0 # i # n " 1.

Definition 4.2.17. — Suppose that q = pn and let r = (r0, . . . , rn"1) be an n-tuple,such that 0 # ri # p " 1, for 0 # i # n " 1. We define &r to be the set of n-tuples(.0, . . . , .n"1), such that

.i = 0, if ri = p " 1 and .i & {0, 1}, otherwise.

We will write 0 = (0, . . . , 0) and 1 = (1, . . . , 1).



Remark 4.2.18. — We hope to prevent some notational confusion. Since we wantLemma 4.2.19 to hold and since dimRU

p"1,Fp= 1, if ri = p " 1, we have to make a

choice for .i, between 0 and 1. We choose 0, since then we can state Lemma 4.2.21in a nice way. However, if ri = p " 1, then

(1 " 0)(p " ri " 1) = (1 " 1)(p " ri " 1) = 0

so it does not matter, whether .i = 0 or .i = 1, and we will exploit this in ournotation. We note that the definition of b! is independent of the set &r and we mighthave ! & &r, !& !& &r, but b! = b!# .

Lemma 4.2.19. — The set {b! : ! & &r} is a basis of RUr .

Proof. — Let r be an integer, such that 0 # r # p" 1. If r = p" 1 , then dim Rr = pand E0 is in RU

r . If 0 # r < p " 1, then dimRr = 2p and E0 and Ep"1"r are twolinearly independent vectors in RU

r .Let r be an n-tuple. Then by above vectors b!, for ! & &r, span a linear subspace

of RUr of dimension |&r|. Also by above, dimRr = |&r|q. Since, U is a p-Sylow

subgroup of "& of order q and by Theorem 4.2.5 Rr is an injective object in Rep!# ,Corollary 4.0.9 implies that

dim RUr = |&r|.

Hence, the set {b! : . & &r} is a basis of RUr .

Lemma 4.2.20. — Let r = (r0, . . . , rn"1) be an n-tuple, with 0 # ri # p " 1, let! = (.0, . . . , .n"1) be an n-tuple such that .i & {0, 1}, for every i, and let b! be avector in RU

r , then the action of H is given by!

, 00 µ

"b! = ,r(,µ)q"1"r(,µ"1)!"(p"r"1)b!

where r = r0 + r1p + · · · + rn"1pn"1 and

! " (p " r " 1) = .0(p " r0 " 1) + .1(p " r1 " 1)p + · · · + .n"1(p " rn"1 " 1)pn"1.

Proof. — This follows from Proposition 4.2.12 and Lemma 4.2.14. We note that theaction on each tensor component is twisted by Fr.

Lemma 4.2.21. — Suppose that q = pn and let r = (r0, . . . , rn"1) be an n-tuple, suchthat 0 # ri # p " 1, for each i. Let b0 be a vector in Rr. Let

r = r0 + r1p + · · · + rn"1pn"1.

Then("b0)Fp

$= Vr,Fp. (det)q"1"r




Proof. — Let Wr be the subspace of Rr given by

Wr = Wr0 . · · ·. Wrn"1

with the notation of the Proposition 4.2.13. We have

0 != ("b0)Fp# Wr.

Proposition 4.2.13 applied to every tensor component implies that

Wr$= Vr,Fp

. (det)q"1"r

which is irreducible. Hence, we must get the whole of Wr.

Corollary 4.2.22. — Let ( : H # F!p and let a and r be unique integers, such

that 1 # a, r # q " 1 and

(++

1 00 &

,,= ,a ', & F!

q , (++

& 00 &"1

,,= ,r ', & F!

q .

Let r = r0 + r1p + · · · + rn"1pn"1, where 0 # ri # p " 1 for each i, and let r =(r0, . . . , rn"1). If ( != (s then

inj "%,! $= Rr . (det)a+r.

If ( = (s theninj "%,! $= Rp"1 . (det)a $= Vp"1,Fp

. (det)a.

Proof. — Lemma 4.2.21 implies the existence of an exact sequence

0 "# Vr,Fp"# Rr . (det)r

of "-representations. It is enough to show that Rr is an indecomposable injectiveobject in Rep!. The rest follows from Propositions 3.2.2 and 4.0.4.

Theorem 4.2.5 says that the restriction of Rr to "& is indecomposable. In particular,Rr must be indecomposable as a "-representation. Moreover, Theorem 4.2.5 saysthat the restrictions of Rr to "& is an injective object in Rep!# . Since U is a p-Sylowsubgroup of both " and "&, Proposition 4.0.7 implies that Rr is an injective object inRep!. Finally, the last isomorphism follows directly from the definition of Rp"1.

4.2.3. Computation of H!-modules. — We will compute the action of Tns

on RUr .

Proposition 4.2.23. — Let q = pn and let r = (r0, . . . , rn"1) be the n-tuple, suchthat 0 # ri # p " 1, for every i. Let ! & &r and let b! be a vector in Rr.

(i) Suppose that for some index j, .j = 0 and rj != p " 1 then%

u$U

un"1s b! = 0.



(ii) Suppose that r != 0. Moreover, suppose that for every i, if .i = 0 then ri = p"1then b! = b1 and %

u$U

un"1s b1 = ("1)1+|r|b0

where |r| = r0 + r1p + · · · + rn"1pn"1.(iii) Suppose that r = 0 and ! = 1, then

%

u$U

un"1s b1 = "(b0 + b1).

This covers all the possible pairs (r, .), such that . & &r.

Remark 4.2.24. — We note that b1 is well defined even if 1 !& &r. See Definitions4.2.16 and 4.2.17.

Proof. — Let r be an integer such that 0 # r # p " 1 and let . & {0, 1} such that. = 0, if r = p " 1. Let E(1"()(p"r"1) be a vector in Rr. We observe that

n"1s E(1"()(p"r"1) = ("1)p"1+((p"r"1)Ep"1+((p"r"1).

If r != p " 1 this follows from Lemma 4.2.9, and if r = p " 1, this follows from theisomorphism Rp"1

$= Vp"1,Fp. Moreover, if . = 0 then Proposition 4.2.13 implies that

ek

k!Ep"1 = 0 ' k > r

and if . = 1 then Lemma 4.2.9 implies that

ek

k!E2p"2"r = 0 ' k > 2p " 2 " r.

Let r be an n-tuple, r = (r0, . . . , rn"1), with 0 # ri # p " 1, let ! & &r. We recallthat

b! = E(1"(0)(p"1"r0) . · · ·. E(1"(n"1)(p"1"rn"1).

Hence we may write %

u$U

un"1s b! = sgnr,(

%

k

'kAk

where the sum is taken over all the n-tuples k = (k0, . . . , kn"1) of non-negativeintegers, moreover

'k =%

&$Fq

,k0+k1p+···+kn"1pn"1

and

Ak =ek0

k0!Ep"1+(0(p"r0"1) . · · ·. ekn"1

kn"1!Ep"1+(n"1(p"rn"1"1)

andsgnr,( = ("1)q"1"!"(p"1"r)

where ! " (p " 1 " r) ='n"1

i=0 .i(p " 1 " ri)pi. We have acted by n"1s on each

tensor component and then expanded the action of u & U on each tensor component



and rearranged the summation. We will show that the sum on the right hand sidevanishes, unless r and ! are of a special form. We will break up the argument intoseveral lemmas.

Lemma 4.2.25. — Let S(r, .) be the subset of the set of all the n-tuples of non-negative integers k = (k0, . . . , kn"1), such that k & S(r, .) if and only if for each i,the following holds:

(i) if .i = 0 then ki = ri,(ii) if .i = 1 then ki = p " 1 or ki = 2p " ri " 2.

Then %

u$U

un"1s b! = sgnr,(

%

k$S(r,()

'kAk.

Proof of Lemma. — If .i = 0 and ki > ri or .i = 1 and ki > 2p" 2" ri then Ak = 0,since the i-th tensor component of Ak vanishes by the argument above. Moreover,Lemma 4.2.9 implies that

eki

ki!Ep"1+(i(p"ri"1) & (Ep"1+(i(p"ri"1)"ki

)Fp.

The vector'

u$U un"1s b! is fixed by U . Since, by Lemma 4.2.19, vectors b!# , for

!& & &r, form a basis of RUr there exist scalars µ(# & Fp such that

%

u$U

un"1s b( =

%

(#$#r

µ(#b(# .

Hence, it is enough to sum over the n-tuples k of non-negative integers such that, foreach i, we have

p " 1 + .i(p " ri " 1) " ki = (1 " .&i)(p " 1 " ri)

for some .&i & {0, 1}. Hence, ki is of the form

ki = p " 1 + (.i " 1)(p " ri " 1) or ki = p " 1 + .i(p " ri " 1).

If .i = 0 and ki is of the form as above then the inequality ki # ri can be fulfilledif and only if ki = ri. If .i = 1, then ki # 2p " ri " 2 implies that ki = p " 1 orki = 2p " ri " 2.

Lemma 4.2.26. — Let k & S(r, .) and let k = k0 + k1p + · · · + kn"1pn"1. Supposethat 'k != 0 then one of the following holds:

(i) r = 0, . = 1 and k = (2(p " 1), . . . , 2(p " 1)) = 2(p " 1),(ii) k = q " 1.

Proof of Lemma. — Since k & S(r, .), for each i we have the inequalities:

0 # ki # 2p " 2 " ri # 2(p " 1).



Hence, 0 # k # 2(q " 1), moreover k = 2(q " 1) if and only if r = 0, . = 1 andk = (2(p " 1), . . . , 2(p " 1)). If k = 0 or k > 0 and q " 1 does not divide k then

'k =%

&$Fq

,k = 0.

We note that 00 = 1 comes from the action by the identity matrix. If k > 0 and q" 1divides k, then

'k =%

&$Fq

,k = "1.

Lemma 4.2.27. — Let k & S(r, .) and let k = k0 + k1p + · · · + kn"1pn"1. Supposethat k = q " 1 then k = (p " 1, . . . , p " 1) = p " 1.

Proof of Lemma. — Since k & S(r, .) we may define integers ai and a&i, such that for

each i,ai + a&

i = ki

and 0 # ai, a&i # p " 1, as follows. If .i = 0, then ai = ri and a&

i = 0. If .i = 1 andki = p" 1, then ai = p" 1 and a&

i = 0. If .i = 1 and ki = 2p" ri " 2, then ai = p" 1and a&

i = p " 1 " ri. Then q " 1 = k implies that

a0 + a1p + · · ·+ an"1pn"1 = (p " 1 " a&

0) + (p " 1 " a&1)p + · · · + (p " 1 " a&

n"1)pn"1.

Since 0 # ai, a&i # p " 1, for every i, this implies that

ai = p " 1 " a&i, ' i.

If .i = 1 and ki = p" 1, then we are done. If .i = 0 then by definition a&i = 0 and by

above ai = p " 1, hence ki = ai + a&i = p " 1. If .i = 1 and ki = 2p " 2 " ri then by

definition ai = p " 1 and by above a&i = 0, hence ki = ai + a&

i = p " 1.

We return to the main body of the proof of Proposition 4.2.23.Suppose that for some index j, we have .j = 0 and rj != p " 1. If 'k != 0 for

some k & S(r, .), then Lemmas 4.2.26 and 4.2.27 imply that either k = p " 1 ork = 2(p " 1). However, the definition of S(r, .) implies that kj = rj < p " 1. Hence'k = 0 for all k & S(r, .) and Lemma 4.2.25 implies that

%

u$U

un"1s b( = 0.

So we obtain part (i) of the Proposition. We note that this case includes r = 0 and! != 1.

Suppose that r != 0. Moreover, suppose that r and . are such that for every i,if .i = 0 then ri = p " 1. Lemmas 4.2.26 and 4.2.27 imply that if k & S(r, .) and'k != 0 then k = p " 1. Lemma 4.2.25 implies that

%

u$U

un"1s b( = sgnr,(("1)Ap"1.



We will compute what happens on each tensor component of Ap"1. If .i = 0, thenby our assumption on r and ., we have ri = p " 1 and

ep"1

(p " 1)!Ep"1 =

!p " 1

0

"E0 = Ep"ri"1.

If .i = 1 then

ep"1

(p " 1)!E2p"2"ri =

!p " 1

0

"Ep"ri"1 = Ep"ri"1.

The above calculation gives us

Ap"1 = Ep"r0"1 . · · ·. Ep"rn"1"1 = b0.

Moreover, if p = 2, then 1 = "1 and if p != 2 then ("1)p"1+(i(p"1"ri) = ("1)ri ,trivially, if .i = 1 and since ri = p " 1 if .i = 0. Hence, sgnr,( = ("1)|r| and

%

u$U

un"1s b! = ("1)|r|+1b0.

We claim that in this case b! = b1. Indeed, if ri != p " 1 then our assumption on rand . implies that .i = 1 and if ri = p " 1, then

(1 " .i)(p " 1 " ri) = (1 " 1)(p " 1 " ri) = 0.

Hence, b( = b1, see 4.2.16. This establishes part (ii) of the Proposition.The only case left is r = 0 and ! = 1. Arguing as before we get that

%

u$U

un"1s b1 = sgn0,1("1)(Ap"1 + A2(p"1)).

We compute what happens on each tensor component of A2(p"1):

e2p"2

(2p " 2)!E2p"2 =

!2p " 2

0

"E0 = E0.

And by Definition 4.2.16, b1 = E0 . · · ·. E0. Since sgn0,1 = 1 we get%

u$U

un"1s b1 = "(b1 + b0).

This establishes part (iii) of the Proposition.

Remark 4.2.28. — We think of .(det)a as a twist, that is, it changes the action,but does not change the underlying vector space. Moreover, since U # "& and ns & "&,Proposition 4.2.23 does not change if we twist the action by (det)a.

Remark 4.2.29. — We know that something like%

u$U

un"1s b1 = ("1)1+|r|b0

has to happen by Lemma 4.1.1.



Lemma 4.2.30. — Let b1 and b0 be vectors in R0. Then

("(b1 + b0))Fp

$= Vp"1,Fp.

Proof. — The vector b1+b0 is fixed by U . Moreover, by Lemma 4.2.20 H acts triviallyon it. By Proposition 4.2.23

(b1 + b0)Tns =%

u$U

un"1s (b1 + b0) = "(b1 + b0).

Hence(b1 + b0)Fp

$= M1,!

as H!-module and Lemma 3.1.8 gives us the result.

Corollary 4.2.31. — Let ( : H # F!p be a character, such that ( = (s and let a

be the unique integer, such that 1 # a # q " 1 and

(++

1 00 &

,,= ,a ', & F!

q

theninj "%,S - inj "%,! $= R0 . (det)a.

Proof. — This is a rerun of the proof of Corollary 4.2.22. Lemma 4.2.21 and Lemma4.2.30 imply the existence of an exact sequence

0 "# V0,Fp- Vp"1,Fp

"# R0

of "-representations. So it is enough to show that R0 is an injective object in Rep!

and that it has at most 2 direct summands. The rest follows from Proposition 4.0.4and Proposition 3.2.2. Theorem 4.2.5 says that the restriction of R0 to "& has exactly2 direct summands, hence R0 may have at most 2 direct summands. Moreover,Theorem 4.2.5 says that the restriction of R0 to "& is an injective object in Rep!# .Since U is a p-Sylow subgroup of " and "& contains U , Proposition 4.0.7 implies thatR0 is an injective object in Rep!.

Definition 4.2.32. — Let / : H # F!p be a character, given by

/ :!

, 00 µ

"*"# ,µ"1.

Lemma 4.2.33. — Suppose that q = pn and let ( : H # F!p be a character. Let r be

the unique integer, such that 0 # r < q " 1 and

(++

& 00 &"1

,,= ,r ', & F!

q .

Let r = r0 + r1p + · · · + rn"1pn"1, where 0 # ri # p " 1 for each i, and let

r = (r0, . . . , rn"1).

Let ! = (.0, . . . , .n"1) be an n-tuple, such that .i & {0, 1} for every i, then

((/!"(p"1"r))s = (/(1"!)"(p"1"r).



Moreover, if r = 0, then we suppose that ! != 0 and ! != 1, then

((/!"(p"1"r))s != (/!"(p"1"r)

where ! " (p " 1" r) ='n"1

i=0 .i(p " ri " 1)pi.

Proof. — Since twisting by s does not a#ect det we may assume that

(++

& 00 µ

,,= ,r ',, µ & F!

q .

Then the first part of the lemma amounts to

µr(µ,"1)!"(p"1"r) = ,r(,µ"1)q"1"r"!"(p"1"r) = ,r(,µ"1)(1"!)"(p"1"r).

For the second part, we observe that the equality holds if and only if

µr+2!"(p"1"r) = ,r+2!"(p"1"r)

for every ,, µ & F!q . Hence, equality holds if and only if

n"1%

i=0

(ri + 2(p " 1 " ri).i)pi 0 0 (mod q " 1).

Since, .i & {0, 1} we have

0 # ri + 2(p " 1 " ri).i # 2(p " 1).

The congruence implies that r + 2! " (p" 1" r) must take values 0, q " 1 or 2(q " 1).The extreme values are obtained if and only if r = 0 and . = 0 or r = 0 and . = 1.By our assumptions, both cases are excluded. If

r + 2! " (p " 1" r) = q " 1

then we rewrite this asn"1%

i=0

(p " 1 " ri).ipi =

n"1%

i=0

(p " 1 " ri)(1 " .i)pi.

Hence, for every i we must have

(p " 1 " ri).i = (p " 1 " ri)(1 " .i).

Since 2.i != 1, for every i, this forces ri = p" 1, for every i, but r < q " 1, hence thiscase is also excluded.

Definition 4.2.34. — Suppose that q = pn and let r = (r0, . . . , rn"1) be an n-tuple,such that 0 # ri # p " 1 for every i. We define

0 & &r

given by 0i = 1 if ri != p " 1 and 0i = 0 if ri = p " 1.We further define &&

r to be a subset of &r given by

&&r = &r\{0, 0}.



Remark 4.2.35. — We note that if p = q or r = (p " 1, . . . , p " 1), then &&r = !

and we always have b) = b1.

Lemma 4.2.36. — Suppose that q = pn and let ( : H # F!p be a character. Let r be

the unique integer, such that 0 # r < q " 1 and

(++

& 00 &"1

,,= ,r ', & F!

q .


r = (r0, . . . , rn"1).

If r = 0 then we consider inj "%,S and if r != 0 we consider inj "%,! as representationsof K on which K1 acts trivially.

Suppose that ! & &&r. If r = 0 then we regard b! and b1"! as vectors in (inj "%,S)I1

via the isomorphism of Corollary 4.2.31. If r != 0 then we regard b! and b1"! asvectors in (inj "%,!)I1 via the isomorphism of Corollary 4.2.22.

Then the action of HK on (b!, b1"!)Fpextends to the action of H, so that

(b!, b1"!)Fp$= M!!

as an H-module, where

+! = +1"! = {(/!"(p"1"r), ((/!"(p"1"r))s}.

Proof. — To ease the notation, let

* = (/!"(p"1"r).

We observe that if b1"! = b0, then ! = 0 and if b1"! = b) then ! = 0. Since ! & &&r

neither of the above can occur.By Lemma 4.2.20 and taking into account the twist by a power of det, I acts on

b! via the character *. By the same argument and Lemma 4.2.33 I acts on b1"! viathe character *s. Hence,

b!e* = b! and b1"!e*s = b1"!.

Moreover, Lemma 4.2.33 says that * != *s. The case r = 0 is not a problem, since! & &&

0 implies that 1" ! & &&0. Since H acts on b! and b1"! by di#erent characters,

they are linearly independent. Proposition 4.2.23 implies that

b!Tns =%

u$I1/K1

un"1s b! = 0 and b1"!Tns =

%

u$I1/K1

un"1s b1"! = 0.

Hence, by Lemma 2.2.2

(b!, b1"!)Fp$= (b!)Fp

- (b1"!)Fp$= M*,! - M*s,! $= M!! |HK

as HK-modules. So we define

b!T" = b1"! and b1"!T" = b!

which gives us the required isomorphism of H-modules.



Proposition 4.2.37. — Suppose that q = pn and let ( : H # F!p be a character,

such that ( = (s. We consider the representation

inj "%,! - inj "%,S

as a representation of K, such that K1 acts trivially. We may extend the action ofHK on

(inj "%,! - inj "%,S)I1

to the action of H, such that (inj "%,! - inj "%,S)I1 as an H-module is isomorphic toa direct sum of 2n"1 supersingular modules of H.

More precisely, for every ! & &0 we consider b! as vectors in

(inj "%,! - inj "%,S)I1

via the isomorphism of Corollary 4.2.31. Then the action of HK can be extended tothe action of H so that

(b0, b0 + b1)Fp

$= M!

where + = {(}. If ! & &&0, then

(b!, b1"!)Fp

$= M!!

where +! = +1"! = {(/!"(p"1), ((/!"(p"1))s}.

Proof. — Since, by Lemma 4.2.19 b! for ! & &0 form a basis of RU0 , the second part

implies the first. Since &&0 = &0\{0,1}, the last part of the Proposition is given by

Lemma 4.2.36.Lemmas 4.2.21 and 4.2.30 imply that

(b0)Fp

$= M%,S , (b1 + b0)Fp

$= M%,!

as an HK-module. Hence, by Lemma 2.2.2

(b0, b1 + b0)Fp

$= M%,S - M%,! $= M! |HK

as HK-modules. Hence, if we define

b0T" = b0 + b1 and (b0 + b1)T" = b0

we get the required isomorphism.

Proposition 4.2.38. — Suppose that q = pn, let ( : H # F!p be a character, such

that ( != (s, and let a and r be unique integers, such that 1 # a, r # q " 1 and

(++

1 00 &

,,= ,a ', & F!

q , (++

& 00 &"1

,,= ,r ', & F!

q .


r = (r0, . . . , rn"1).

Theninj "%,! - inj "%s,! $= Rr . (det)a+r - Rp"1"r . (det)a

where p " 1 " r = (p " 1 " r0, . . . , p " 1 " rn"1).



We regard the representation inj "%,!- inj"%s,! as a representation of K, on whichK1 acts trivially. Let c and d be the cardinality of the sets:

c = |{ri : ri != p " 1}| and d = |{ri : ri != 0}|

then we may extend the action of HK on

(inj "%,! - inj "%s,!)I1

to the action of H, such that (inj "%,! - inj "%s,!)I1 as an H-module is isomorphic toa direct sum of L! and 2c"1 + 2d"1 " 2 supersingular modules of H.

More precisely, let b!, for ! & &r, be a basis of (inj "%,!)I1 and let b!, for ! &&p"1"r, be a basis of (inj "%s,!)I1 via the isomorphism above. Then the action of HK

can be extended to the action of H so that

(b0, b1, b0, b1)Fp

$= L!

and(b0, b0)Fp

$= M!

where + = {(, (s}. If ! & &&r, then

(b!, b1"!)Fp$= M!!

where +! = +1"! = {(/!"(p"1"r), ((/!"(p"1"r))s}. If ! & &&p"1"r then

(b!, b1"!)Fp

$= M!!

where +! = +1"! = {(s/!"r, ((s/!"r)s}.

Proof. — The first part of the Proposition follows from Corollary 4.2.22 and Corollary3.2.3. For the second part we observe that since ( != (s, we have r != q" 1 and hencevectors b0, b1, b0 and b1 are linearly independent. Lemma 4.2.21 implies that

(b0)Fp

$= M%,! and (b0)Fp

$= M%s,!

as HK-modules. Lemma 4.2.20 with the appropriate twist by a power of det says thatH acts on b1 by a character (/1"(p"1"r) and H acts on b1 by a character (s/1"r.Lemma 4.2.33 implies that

(/1"(p"1"r) = (s and (s/1"r = (.

Hence,b1e%s = b1 and b1e% = b1.

Proposition 4.2.23 implies that

("1)r+1b1Tns = ("1)r+1%

u$I1/K1

un"1s b1 = b0

and("1)q"rb1Tns = ("1)q"r

%

u$I1/K1

un"1s b1 = b0.



Hence, by Lemma 2.2.2(b0, b1, b0, b1)Fp

$= L! |HK

as HK-modules. We note that if p = 2 then 1 = "1 and if p != 2 then ("1)q"r =("1)r+1. So if we define

b1T" = b1, b1T" = b1, b0T" = b0, b0T" = b0

we get the required isomorphism of H-modules. Moreover,

(b0, b0)Fp

$= M!

as H-module. The last part of the Proposition follows from Lemma 4.2.36. Sincedim(inj "%,!)I1 = 2c and dim(inj "%s,!)I1 = 2d an easy calculation gives us the numberof indecomposable summands.

Remark 4.2.39. — If p = q, then &&r = ! and Propositions 4.2.37 and 4.2.38 spe-

cialise to Proposition 4.1.9.

The following Proposition can be seen as a consolation for the Remark 4.1.6.

Proposition 4.2.40. — Suppose that q = pn, ( != (s and let " be a representationof ", such that "U $= M%,! - M%s,! as an H!-module, and " = (""U )Fp

, then

" $= "%,! - "%s,!.

Proof. — If " is a semi-simple representation of ", then Corollary 3.1.3 implies theLemma. Suppose that " is not semi-simple. Let soc(") be the maximal semi-simplesubrepresentation of ". Since " is generated by "U as a "-representation, the space(soc("))U is one dimensional, and hence soc(") is an irreducible representation of ".By Corollary 3.1.3 and symmetry we may assume that

soc(") $= "%,!.

Since, soc(") is irreducible, " is an essential extension of "%,!. By this we mean thatevery non-zero "-invariant subspace of " intersects "%,! non-trivially. This impliesthat there exists an exact sequence

0 "# " "# inj "%,!

of "-representations. After twisting by a power of determinant we may assume that( is given by (

++& 00 µ

,,= ,r , where 0 < r < q " 1. The inequalities are strict,

since ( != (s. Let r be the corresponding n-tuple. Let . & &r and b( & (inj "%,!)U ,then H acts on b( by the character (/("(p"1"r). In particular, if .& & &r, such that.& != ., then H acts on b( and b(# by di#erent characters. As a consequence of this,the submodule M%s,! of "U must be mapped to some subspace (b!)Fp

of (inj "%,!)U ,where . & &r. By examining the action of H , we get that (s = (/!"(p"1"r). Thisimplies that

. " (p " 1 " r) + r 0 0 (mod q " 1).



Since 0 < r < q " 1 and . & &r, we have

0 < . " (p " 1" r) + r # q " 1.

Hence, we get an equality on the right hand side, which implies that, for each i,(1 " .i)(p " 1 " ri) = 0. So . = 0, and b( = b1, see 4.2.34 and 4.2.35. However, byProposition 4.2.23 (ii)

b1Tns = ("1)r+1b0 != 0.

We obtain a contradiction, since Tns kills M%s,!.


CHAPTER 5

COEFFICIENT SYSTEMS

We closely follow [13] and [14, §V], where the G-equivariant coe!cient systems ofC-vector spaces are treated. In fact, the results of this Section do not depend on theunderlying field. Our motivation to use coe!cient systems stems from [12], wherethe equivariant coe!cient systems of Fp-vector spaces of finite Chevalley groups areconsidered.

5.1. Definitions

The Bruhat-Tits tree X of G is the simplicial complex, whose vertices are thesimilarity classes [L] of oF -lattices in a 2-dimensional F -vector space V and whoseedges are 1-simplices, given by families {[L0], [L1]} of similarity classes such that

!F L0 2 L1 2 L0.

We denote by X0 the set of all vertices and by X1 the set of all edges.

Definition 5.1.1. — Let & be a simplex in X , then we define

K(&) = {g & AutF (V ) : g& = &}.

By fixing a basis {v1, v2} of V we identify G with AutF (V ). Let

&0 = [oF v1 + oF v2] and &1 = {[oF v1 + oF v2], [oF v1 + pF v2]}.

Then &0 is a vertex and K(&0) = F!K, and &1 is an edge containing a vertex &0.Moreover, K(&1) is the group generated by I and $.

Definition 5.1.2. — A coe!cient system V (of Fp-vector spaces) on X consists of

– Fp vector spaces V# for each simplex & of X , and– linear maps r##

# : V## # V# for each pair & + && of simplices of X such that forevery simplex &, r#

# = idV! .

52 CHAPTER 5. COEFFICIENT SYSTEMS

Definition 5.1.3. — We say the group G acts on the coe!cient system V , if forevery g & G and for every simplex & there is given a linear map

g# : V# "# Vg# ,

such that– gh# , h# = (gh)#, for every g, h & G and for every simplex &,– 1# = idV! for every simplex &,– the following diagram commutes for every g & G and every pair of simplices

& + &&:

V#g#

!! Vg#

V##

r##

#

$$

g##!! Vg##

rg##

g#

$$.

In particular, the stabiliser K(&) acts linearly on V# for any simplex &.

Definition 5.1.4. — A G-equivariant coe!cient system (V+ )+ on X is a coe!cientsystem on X together with a G-action , such that the action of the stabiliser K(&) ofa simplex & on V# is smooth.

Remark 5.1.5. — The definition given in [14] §V, requires the action to factorthrough a discrete quotient.

Let COEFG denote the category of all equivariant coe!cient systems on X ,equipped with the obvious morphisms.

The following observation will turn out to be very useful. Suppose that G acts ona coe!cient system V = (V#)#. Let 1 & be an edge containing a vertex 1 . There existsg & G, such that 1 & = g&1 and 1 = g&0. Then

V+ = g#0V#0 , V+ # = g#1V#1

andr+ #

+ = g#0 , r#1#0

, (g"1)+ # .

5.2. Homology

Let X(0) be the set of vertices on the tree and let X(1) be the set of oriented edgeson the tree. We will say that two vertices & and && are neighbours if {&, &&} is anedge. And we will write

(&, &&)to mean a directed edge going from & to &&. Let V = (V+ )+ be an equivariant coe!cientsystem. We define a space of oriented 0-chains to be

Corc (X(0),V) = Fp-vector space of all maps 2 : X(0) "#

&#$X0

V#


5.3. SOME COMPUTATIONS OF H0(X, V) 53

such that– 2 has finite support and– 2(&) & V# for every vertex &.

Similarly, the space of oriented 1-chains is

Corc (X(1),V) = Fp-vector space of all maps 2 : X(1) "#

&{#,##}$X1

V{#,##}

such that– 2 has finite support,– 2((&, &&)) & V{#,##} ,– 2((&&, &)) = "2((&, &&)) for every oriented edge (&, &&).

The group G acts on Corc (X(0),V) via

(g2)(&) = gg"1#(2(g"1&))

and on Corc (X(1),V) via

(g2)((&, &&)) = g{g"1#,g"1##}(2((g"1&, g"1&&))).

The action on both spaces is smooth.The boundary map is given by

3 : Corc (X(1),V) "# Cor

c (X(0),V)

2 *"#.& *#

%

##

r{#,##}# (2((&, &&)))

/

where the sum is taken over all the neighbours of &. The map 3 is G-equivariant.We define H0(X,V) to be the cokernel of 3. It is naturally a smooth representation

of G.

5.3. Some computations of H0(X,V)

Throughout this section we fix an equivariant coe!cient system V = (V+ )+ , withthe restriction maps given by r+ #

+ . Our first lemma follows immediately from thedefinition of 3.

Lemma 5.3.1. — Let 2 be an oriented 1-chain supported on a single edge 1 = {&, &&}.Let

v = 2((&, &&)).

Then3(2) = 2# " 2## ,

where 2# and 2## are 0-chains supported only on & and && respectively. Moreover,

2#(&) = r+#(v) and 2##(&&) = r+

## (v).



Lemma 5.3.2. — Let 2 be a 0-chain supported on a single vertex &. Suppose that therestriction map r#1

#0is an injection, then the image of 2 in H0(X,V) is non-zero.

Proof. — Since every restriction map is conjugate to r#1#0

by some element of G, itfollows that every restriction map is injective.

Let 2& be a non-zero oriented 1-chain. We may think of the support of 2& as theunion of edges of a finite subgraph T of X . Since all the restriction maps are injective,Lemma 5.3.1 implies that 3(2&) will not vanish on the vertices of T , which have onlyone neighbour in T . In particular, 3(2&) will be supported on at least 2 vertices.Hence, 2 !& 3Cor

c (X(1),V).

Lemma 5.3.3. — Let 2 be 0-chain. Suppose that the restriction map r#1#0

is surjective,then there exists a 0-chain 20, supported on a single vertex &0, such that

2 + 3Corc (X(1),V) = 20 + 3Cor

c (X(1),V).

Proof. — Since every restriction map is conjugate to r#1#0

by some element of G, itfollows that every restriction map is surjective.

It is enough to prove the statement when 2 is supported on a single vertex 1 , sincean arbitrary 0-chain is a finite sum of such. If 1 = &0 then we are done. Otherwise,there exists a directed path going from &0 to 1 , consisting of finitely many directededges (&0, 11), . . . , (1m, 1).

We argue by induction on m. Let v = 2(1). Since r{+m,+}+ is surjective there exists

v& & V{+m,+}, such thatr{+m,+}+ (v&) = v.

Let 2& be an oriented 1-chain supported on the single edge {1m, 1} with 2&((1m, 1)) =v&. By Lemma 5.3.1 2+3(2&) is supported on a single vertex 1m. Since, the number ofedges in the directed path has decreased by one, the claim follows from induction.

The following special case will be used in the calculations of modules of the Heckealgebra.

Lemma 5.3.4. — Let 20 be a 0-chain supported on a single vertex &0. Let

v0 = 20(&0)

and suppose that there exists v1 & V#1 , such that

r#1#0

(v1) = v0.

Let 2& be a 0-chain supported on a single vertex &0 with

2&(&0) = r#1#0

(($"1)#1 (v1)),

then$"120 + 3Cor

c (X(1),V) = 2& + 3Corc (X(1),V).


5.3. SOME COMPUTATIONS OF H0(X, V) 55

Proof. — We observe that $&0 = $"1&0 and &1 = {&0, $&0}. The 0-chain $"120 issupported on a single vertex $&0 with

($"120)($&0) = ($"1)#0 (v0).

Let 21 be an oriented 1-chain supported on a single edge &1 with

21((&0, $&0)) = ($"1)#1(v1).

From Lemma 5.3.1 we know that 3(21) is supported only on &0 and $&0. Moreover,

3(21)($&0) = r#1"#0

(21(($&0, &0)) = r#1"#0

("($"1)#1 (v1))

= "(r#1"#0

, ($"1)#1)(v1) = "(($"1)#0 , r#1#0

)(v1) = "($"1)#0 (v0)

and3(21)(&0) = r#1

#0(21((&0, $&0))) = r#1

#0(($"1)#1(v1)).

Hence3(21) = 2& " $"120

and that establishes the claim.

Proposition 5.3.5. — Suppose that the restriction map r#1#0

is an isomorphism ofvector spaces. Then

H0(X,V)|K(#0)$= V#0

andH0(X,V)|K(#1)

$= V#1 .

Moreover, the diagram

V#0

$= !! H0(X,V)

V#1

r#1#0

$$

$= !! H0(X,V)

id$$

of F!I-representations commutes.

Proof. — Let Corc (&0,V) be a subspace of Cor

c (X(0),V) consisting of the 0-chainswhose support lies in the simplex &0, with the understanding that the 0-chain whichvanishes on every simplex is supported on the empty simplex. Let 4 be the composition

4 : Corc (&0,V) %"# Cor

c (X(0),V) "# H0(X,V).

Then 4 is K(&0) equivariant. Moreover, Lemma 5.3.2 says that 4 is an injection andLemma 5.3.3 says that it is a surjection. Hence

4 : Corc (&0,V) $= H0(X,V)|K(#0).

Let ev0 be the map

ev0 : Corc (&0,V) "# V#0

2 *"# 2(&0)



then ev0 is an isomorphism of K(&0)-representations. Hence

4 , (ev0)"1 : V#0$= H0(X,V)|K(#0).

Since V is G-equivariant, the map r#1#0

is F!I = K(&1) % K(&0)-equivariant and sinceit is isomorphism of vector spaces, we obtain that

4 , (ev0)"1 , r#1#0

: V#1 |F!I$= H0(X,V)|F!I .

We claim that this isomorphism is in fact K(&1)-equivariant. Let v1 & V#1 , let v0 =r#1#0

(v1) and let 20 & Corc (&0,V), such that 20(&0) = v0. Then

(4 , (ev0)"1 , r#1#0

)(v1) = 20 + 3Corc (X(1),V).

By Lemma 5.3.4

$"120 + 3Corc (X(1),V) = 2& + 3Cor

c (X(1),V),

where 2& & Corc (&0,V) with 2&(&0) = r#1

#0(($"1)#1(v1)). This implies that

$"1(4 , (ev0)"1 , r#1#0

)(v1) = (4 , (ev0)"1 , r#1#0

)(($"1)#1(v1)).

Since $"1 and F!I generate K(&1) this proves the claim.The commutativity of the diagram follows from the way the isomorphisms are

constructed.

5.4. Constant functor

The content of this Section is essentially [12] Lemma 1.1 and Theorem 1.2. LetRepG be the category of smooth Fp-representations of G. Let # be a smooth repre-sentation of G with the underlying vector space W . Let & be a simplex on the tree X ,we set

(K")# = W .

If & and && are two simplices, such that & + && then we define the restriction map

r##

# = idW .

For every g & G and every simplex & in X we define a linear map g# by

g# : (K")# "# (K")g#, v *"# #(g)v.

This gives a G-equivariant coe!cient system on X , which we denote by K".

Definition 5.4.1. — We define the constant functor

K : RepG "# COEFG, # *"# K" .

Lemma 5.4.2. — Let # be a smooth representation of G, then

H0(X,K") $= #

as a G-representation.


5.5. DIAGRAMS 57

Proof. — We have an evaluation map

ev : Corc (X(0),K") "# #, 2 *"#

%

#$X(0)

2(&).

Since the restriction maps are just idW , Lemma 5.3.1 implies that the image of theboundary map 3Cor

c (X(1),K") is contained in the kernel of ev. Hence, we get aG-equivariant map

H0(X,K") "# #.

It is enough to show that this is an isomorphism of vector spaces, and this is impliedby Proposition 5.3.5.

Proposition 5.4.3. — Let V = (V#)# be a G-equivariant coe!cient system with therestriction maps r##

# and let (#,W) be a smooth representation of G, then

HomCOEFG(V ,K") $= HomG(H0(X,V), #).

Proof. — Any morphism of G-equivariant coe!cient systems will induce a G-equivariant homomorphism in the 0-th homology. Hence by Lemma 5.4.2 we have amap

HomCOEFG(V ,K") "# HomG(H0(X,V), #).

We will construct an inverse of this. Let ' & HomG(H0(X,V), #), let & be a vertexon the tree X , let v be a vector in V#, and let 2#,v be a 0-chain, such that

Supp2#,v + &, 2#,v(&) = v,

then we define

'# : V# "# W , v *"# '(2#,v + 3Corc (X(1),V)).

Let 1 be an edge in X with vertices & and &&, we define

'+ : V+ "# W , v *"# '#(r+#(v)).

Lemma 5.3.1 implies that the definition of '+ does not depend on the choice of vertex.Hence, the collection of linear maps ('#)# is a morphism of coe!cient systems, whichinduces ' on the 0-th homology. An easy check shows that ('#)# respect the G-actionon V and K".

5.5. Diagrams

Definition 5.5.1. — Let DIAG be the category, whose objects are diagrams

D0

D1

r

$$



where ("0, D0) is a a smooth Fp-representation of K(&0), ("1, D1) is a smooth Fp-representation of K(&1), and r & HomF!I(D1, D0).

The morphisms between two objects (D0, D1, r) and (D&0, D

&1, r

&) are pairs (*0, *1),such that *0 & HomK(#0)(D0, D&

0), *1 & HomK(#1)(D1, D&1) and the diagram:

D0*0

!! D&0

D1

r

$$

*1!! D&

1

r&$$

of F!I representations commutes.

The main result of this section is Theorem 5.5.4, which says that the categoriesDIAG and COEFG are equivalent. It is easier to work with objects of DIAG thanthe coe!cient systems.

Definition 5.5.2. — Let V = (V#)# be an object in COEFG. Let D : COEFG #DIAG be a functor, given by

V *"#

V#0

V#1

r#1#0

$$

.

We will construct a functor C : DIAG # COEFG and show that the functorsC and D induce an equivalence of categories. The key point here is that G actstransitively on the vertices of X .

5.5.1. Underlying vector spaces. — Let D = (D0, D1, r) be an object in DIAG.Let i & {0, 1}, we define c-IndG

K(#i) "i, to be a representation of G whose underlyingvector space consists of functions

f : G "# Di

such thatf(kg) = "i(k)f(g) ' g & G, ' k & K(&i)

and Supp f is compact modulo the centre. The group G acts by the right translations,that is

(gf)(g1) = f(g1g).Let 1 be a vertex on the tree X , then there exists g & G, such that 1 = g&0. Let

F+ = {f & c-IndGK(#0) "0 : Supp f + K(&0)g"1}.

The space F+ is independent of the choice of g. Let 1 & be an edge on the tree X , thenthere exists g & G such that 1 & = g&1. We define

F+ # = {f & c-IndGK(#1) "1 : Supp f + K(&1)g"1}.

We observe that F+ # is also independent of the choice of g.


5.5. DIAGRAMS 59

5.5.2. Restriction maps. — Let i & {0, 1}, then F#i is naturally isomorphic toDi as a K(&i) representation. The isomorphism is given by

evi : F#i "# Di, f *"# f(1).

The inverse is given by

ev"1i : Di "# F#i , v *"# fv

where fv(k) = "i(k)v, if k & K(&i), and 0 otherwise. Let

r#1#0

= ev"10 ,r , ev1 .

Then r#1#0

is an F!I-equivariant map from F#1 to F#0 . If v & D1 then it sends

r#1#0

: fv *"# fr(v).

We observe, for the purposes of Theorem 5.5.4, that

#D = (F#0 ,F#1 , r#1#0

)

is an object of DIAG. Moreover, ev = (ev0, ev1) is an isomorphism of diagramsbetween D and #D. We will show later on that ev induces a natural transformationbetween certain functors.

Let 1 & be an edge containing a vertex 1 , then there exists g & G, such that 1 = g&0

and 1 & = g&1. Moreover, g can only be replaced by gk, where k & K(&0) % K(&1) =F!I. We define

r+ #

+ : F+ # "# F+ , f *"# gr#1#0

(g"1f)

where g acts on the space c-IndGK(#0) D0 and g"1 on the space c-IndG

K(#1) D1. Since, ris F!I-equivariant we have

"0(k) , r#1#0

, "1(k"1) = r#1#0

for all k & F!I. Hence, the map r+ #

+ is independent of the choice of g. Explicitly, letv = f(g"1), then

r+ #

+ : f *"# gfr(v).

Let 1 be any simplex then we define the map r++ = idF" .

5.5.3. G-action. — So far from a diagram we have constructed a coe!cient system.We need to show that G acts on it. Let i & {0, 1} and let f & c-IndG

K(#i) Di. For anyg & G we have

Supp(gf) = (Supp f)g"1.

Hence for any simplex 1 we obtain a linear map

g+ : F+ "# Fg+ , f *"# gf.



Moreover, 1+ = idF" and gh+ , h+ = (gh)+ , for any g, h & G. Let 1 & be an edgecontaining a vertex 1 . We need to show that the diagram:

F+g+

!! Fg+

F+ #

r+ #

+

$$

g+ #!! Fg+ #

rg+ #

g+

$$

commutes. There exists g1 & G such that 1 = g1&0 and 1 & = g1&1. Moreover, such g1

is determined up to a multiple g1k, where k & F!I. Let f & F+ # and let v = f(g"11 ),

thenr+ #

+ (f) = g1fr(v).

Hence(g+ , r+ #

+ )(f) = gg1fr(v).

Since g1 & = gg1&1, g1 = gg1&0 and (gf)((gg1)"1) = f(g"11 ) = v we obtain

(rg+ #

g+ , g+ #)(f) = rg+ #

g+ (gg1fv) = gg1fr(v).

Hence the diagram commutes.

5.5.4. Morphisms. — Let D& = (D&0, D

&1, r

&) be another diagram, let * = (*0, *1)be a morphism of diagrams

* : D "# D&

and let F & = (F &+ )+ be a coe!cient system associated to D& via the construction

above. Let 1 be any simplex on the tree. If 1 is a vertex let i = 0 and if 1 is an edge,let i = 1. There exists some g & G such that 1 = g&i. Let f & V+ and let v = f(g"1)we define a map

*+ : F+ "# F &+ , f *"# gf*i(v)

where f*i(v) is the unique function in F &#i

, such that f*i(v)(1) = *i(v). Since the map*i is K(&i)-equivariant, *+ is independent of the choice of g.

We will show that the maps (*+ )+ are compatible with the restriction maps. Let 1 &

be an edge containing a vertex 1 . We claim that the diagram

F+*+

!! F &+

F+ #

r+ #

+

$$

*+ #!! F &

+ #

(r&)+ #

+

$$

commutes. There exists g & G such that 1 = g&0 and 1 & = g&1. Let f & F+ # and letv = f(g"1). Then

(*+ , r+ #

+ )(f) = *+ (gfr(v)) = gf*0(r(v))

and((r&)+ #

+ , *+ #)(f) = (r&)+ #

+ (gf*1(v)) = gfr#(*1(v)).


5.5. DIAGRAMS 61

Since (*0, *1) is a morphism of diagrams

*0(r(v)) = r&(*1(v)).

Hence the diagram commutes as claimed and (*+ )+ are compatible with the restrictionmaps.

Finally, we will show that the maps (*+ )+ are compatible with the G-action. Let 1be any simplex on the tree. To ease the notation, for every h & G we denote by h+

the action of h on both (F+ )+ and (F &+ )+ . Let 1 be a simplex on the tree X and let

h & G. We claim that the diagram

Fh+*h+

!! F &h+

F+

h+

$$

*+!! F &

+

h+

$$

commutes. If 1 is an edge let i = 1, if 1 is a vertex let i = 0. There exists g & G, suchthat 1 = g&i. Let f & F+ and let v = f(g"1), then

*h+ (h+ (f)) = *h+ (hgfv) = hgf*i(v)

and

h+ (*+ (f)) = h+ (gf*i(v)) = hgf*i(v).

Hence, the diagram commutes as claimed and the collection (*+ )+ defines a morphismof equivariant coe!cient systems.

5.5.5. Equivalence

Definition 5.5.3. — Let C be a functor

C : DIAG "# COEFG

which sends a diagram D to the coe!cient system (F+ )+ as above.

One needs to check that given three diagrams and two morphisms between them

D*""# D& *&

"""# D&&

we have

C(*& , *) = C(*&) , C(*).

However, that is immediate from the construction of C(*) in Section 5.5.4.

Theorem 5.5.4. — The functors C and D induce an equivalence of categories be-tween DIAG and COEFG.



Proof. — Let D = (D0, D1, r) be an object in DIAG. Then

(D , C)(D) = #D = (F#0 ,F#1 , r#1#0

)

with the notation of Section 5.5.2. The isomorphism

ev : #D $= D

of Section 5.5.2 is given by the evaluation at 1. We claim that it induces an isomor-phism of functors between D , C and idDIAG. We only need to check what happensto morphisms. Let D& = (D&

0, D&1, r

&) be another object in the category of diagramsand let * = (*0, *1) be a morphism

* : D "# D&.

Let (D , C)(D&) = #D& = (F &#0

,F &#1

, (r&)#1#0

) and let

(D , C)(*) = #* = ( #*0, #*1)

be a morphism induced by a functor D , C. We need to show that the diagram:

#D& ev !! D&

#D

#*$$

ev !! D

*

$$

commutes. Let i & {0, 1}, let f & F#i and let v = f(1) then

(*i , evi)(f) = *i(v).

From Section 5.5.4 #*i(f) is the unique function in F &#i

, taking value *i(v) at 1. Hence

(evi , #*i)(f) = *i(v).

This implies that the diagram commutes.Conversely, we need to show that the functor C ,D is isomorphic to idCOEFG . Let

V = (V+ )+ be a G-equivariant coe!cient system with the restriction maps t+#

+ . ThenD(V) is a diagram given by:

V#0

V#1

t#1#0

$$.

Let k & K(&0) then it acts on V#0 by a linear map k#0 . Similarly, if k & K(&1) then itacts on on V#1 by a linear map k#1 . Let

(C ,D)(V) = F = (F+ )+

with the restriction maps r+ #

+ . We will construct a canonical isomorphism ev = (ev+ )+

ev : F $= V


5.5. DIAGRAMS 63

of G equivariant coe!cient systems. Let 1 be a simplex on the tree. If 1 is a vertexlet i = 0 and if 1 is an edge let i = 1. There exists g & G such that 1 = g&i. Forf & F+ we let v = f(g"1). Then v is a vector in V#i . We define a map ev+ , by

ev+ : F+ "# V+ , f *"# g#iv

where g#i is the linear map coming from the G action on V . If we replace g by gk,for some k & K(&i), then

(gk)#i(f((gk)"1)) = (g#i , k#i , k"1#i

)(f(g"1)) = g#i(f(g"1)).

Hence, the map ev+ is independent of the choice of g. Moreover, ev+ is an isomorphismof vector spaces with the inverse given as follows. Let w & V+ , let v = (g"1)+w, thenv is a vector in W#i . Let fv be the unique function in F+ such that fv(1) = v. Then(ev+ )"1 is given by

(ev+ )"1 : V+ "# F+ , w *"# gfv

where the action by g is on the space c-IndGK(#i) V#i .

The collection of maps (ev+ )+ is G-equivariant. Let h & G, then hf belongs to thespace Fh+ and

evh+ (hf) = (hg)#i((hf)((hg)"1)) = (h+ , g#i)(f(g"1)) = h+ (ev+ (f)).

We need to show that the maps ev+ are compatible with the restriction maps. Let1 & be an edge containing a vertex 1 . We need to show that the diagram

F+ev+ !! V+

F+ #

r+ #

+

$$

ev+ #!! V+ #

t+#

+

$$

commutes. There exists g & G such that 1 = g&0 and 1 & = g&1. Let f be a functionin F+ # . Let v1 = f(g"1) , then v1 is a vector in V#1 . Let v0 = t#1

#0(v1). Then r+ #

+ (f) isthe unique function of F+ taking value v0 at g"1. Hence

(ev+ ,r+ #

+ )(f) = g#0v0.

On the other hand(t+

#

+ , ev+ #)(f) = t+#

+ (g#1v1).

The action of G on V respects the restriction maps, in the sense that the diagram:

V#0

g#0!! V+

V#1

t#1#0

$$

g#1 !! V+ #

t+#

+

$$.

commutes. Hence,t+

#

+ (g#1v1) = g#0v0.



Hence our original diagram commutes and ev = (ev+ )+ defines an isomorphism ofG-equivariant coe!cient systems.

In order to show that the morphism ev induces an isomorphism of functors betweenC , D and idCOEFG we need to check what happens to the morphisms. However theproof is almost identical to the one given for DIAG so we omit it.

Corollary 5.5.5. — Let ("0, V0) be a smooth representation of K(&0) and ("1, V1)a smooth representation of K(&1). Suppose that there exists an F!I-equivariant iso-morphism

r : V1$= V0,

then there exists a unique (up to isomorphism) smooth representation # of G, suchthat

#|K(#0)$= "0, #|K(#1)

$= "1

and the diagram

V0

$= !! #

V1

r

$$

$= !! #

id

$$

of F!I-representations commutes.

Proof. — Let D be the object in DIAG, given by D = (V0, V1, r). Let C(D) be a coef-ficient system corresponding to D, with the restriction maps r+ #

+ . Since (D,C)(D) $= Dand r is an isomorphism, the map r#1

#0is an isomorphism and Proposition 5.3.4 implies

that H0(X, C(D)) satisfies the conditions of the Corollary.The statement of the Corollary can be rephrased as follows: there exists a unique

up to isomorphism smooth representation # of G, such that

D $= D(K").

If #& was another such, then

D(K"# ) $= D $= D(K").

Hence, by Theorem 5.5.4K"# $= K" .

Lemma 5.4.2 implies that

#& $= H0(X,K"#) $= H0(X,K") $= #

and we obtain uniqueness.

Remark 5.5.6. — Let 4W be a subgroup of G generated by s and $. The Iwahoridecomposition says that G = I4WI. Let # be a representation constructed as above,v & # and g & G. Then gv may be determined by decomposing g = u1wu2, whereu1, u2 & I, w & 4W , and then chasing around the diagram.


5.5. DIAGRAMS 65

The simplest example illustrating 5.5.5 is the trivial diagram #1 = (1,1, id). Theproof of Corollary 3.1.9 can be reinterpreted as a construction of a morphism #1 %#D(K"). This gives us an injection of G representations

1 $= H0(X, C(#1)) %"# H0(X,K") $= #.


CHAPTER 6

SUPERSINGULAR REPRESENTATIONS

6.1. Coe!cient systems V!

Let ( : H # F!p be a character, and let "%,J be an irreducible representation of ",

with the notations of Section 3. We consider ( as a character of I and "%,J as arepresentation of K, via

K "# K/K1$= " and I "# I/I1

$= H.

Let #"%,J be the extension of "%,J to F!K such that our fixed uniformiser !F actstrivially, and let #( be the extension of ( to F!I, such that !F acts trivially. The spaceof I1-invariants of #"%,J is one dimensional and F!I acts on it via the character #(. Wefix a vector v%,J such that

"I1%,J = (v%,J )Fp

.

Lemma 6.1.1. — There exists a unique action of K(&1) on (#"%,J -#"%s,J )I1 , extendingthe action of F!I, such that

$"1v%,J = v%s,J and $"1v%s,J = v%,J .

Moreover, with this action

(#"%,J - #"%s,J)I1 $= IndK(#1)F!I #(

as K(&1)-representations.

Proof. — We note that if t & T is a diagonal matrix then $t$"1 = sts, hence(#()" $= 4(s as representations of F!I and Mackey’s decomposition gives us

(IndK(#1)F!I #()|F!I

$= #( - 4(s.

Since(#"%,J - #"%s,J)I1 $= #( - 4(s

68 CHAPTER 6. SUPERSINGULAR REPRESENTATIONS

as F!I-representation, we can extend the action. Explicitly, we consider f &IndK(#1)

F!I #(, such that Supp f = F!I and f(g) = #((g), for all g & F!I. Then the map

f *"# v%,J , $"1f *"# v%s,J

induces the required isomorphism. Since, $ and F!I generate K(&1) the action isunique.

Definition 6.1.2. — Let ( : H # F!p be a character, and let + = {(, (s} we define

D! to be an object in DIAG, given by

#"%,J - #"%s,J

(#"%,J - #"%s,J )I1

$$

where the action of K(&1) on (#"%,J - #"%s,J)I1 is given by Lemma 6.1.1. Moreover, wedefine V! to be a coe!cient system, given by

V! = C(D!).

Lemma 6.1.3. — The diagram D! is independent up to isomorphism of the choicesmade for v%,J and v%s,J .

Proof. — Suppose that instead we choose vectors v&%,J and v&%s,J

and let D&! be the

corresponding diagram. Since, the spaces "I1%,J and "I1

%s,Jare one dimensional there

exist ,, µ & F!p , such that

,v%,J = v&%,J , µv%s,J = v&%s,J

.

The isomorphism, id-µ id : #"%,J - #"%s,J "# #"%,J - #"%s,J

induces an isomorphism of diagrams D!$= D&

! .

Since D! and D(V!) are canonically isomorphic, to ease the notation, we identifythem. Let 2%,J , 2%s,J & Cor

c (X(0),V!) supported on a single vertex &0, such that

2%,J(&0) = v%,J and 2%s,J(&0) = v%s,J .

Let2%,J = 2%,J + 3Cor

c (X(1),V!) and 2%s,J = 2%s,J + 3Corc (X(1),V!)

be their images in H0(X,V!).

Lemma 6.1.4. — We have

(2%,J , 2%s,J)Fp

$= M!

as right H-modules.


6.1. COEFFICIENT SYSTEMS V# 69

Proof. — Since the restriction maps in V! are injective, Lemma 5.3.2 says that 2%,J

and 2%s,J are non-zero. We have

(v%,J )Fp= (#"%,J)I1 $= M%,J and (v%s,J)Fp

= (#"%s,J)I1 $= M%s,J

as HK-modules. Hence 2%,J and 2%s,J are fixed by I1 and

(2%,J)Fp- (2%s,J)Fp

$= M%,J - M%s,J

as HK-modules. Corollary 2.0.7 and Lemma 5.3.4 imply that

2%,JT" = $"12%,J = 2%s,J and 2%s,JT" = $"12%s,J = 2%,J .

Hence(2%,J , 2%s,J)Fp

$= M!

as H-modules.

Lemma 6.1.5. — The vector 2%,J (resp. 2%s,J) generates H0(X,V!) as a G-representation.

Proof. — Lemma 5.3.4 implies that $"12%,J = 2%s,J . Hence, it is enough to showthat 2%,J and 2%s,J generate Cor

c (X(0),V!) as a G-representation. Since, "%,J and"%s,J are irreducible K-representations, 2%,J and 2%s,J will generate the space

Corc (&0,V!) = {2 & Cor

c (X(0),V!) : Supp2 + &0}

as a K-representation. Since the action of G on the vertices of X is transitive, thespace Cor

c (&0,V!) will generate Corc (X(0),V!) as a G-representation.

Corollary 6.1.6. — Let # be a non-zero irreducible quotient of H0(X,V!), then #is a supersingular representation.

Proof. — Lemma 6.1.5 implies that the images of 2%,J and 2%s,J in # are non-zero.Hence, by Lemma 6.1.4, #I1 will contain a supersingular module M! , then Corollary2.1.5 implies that # is supersingular.

Proposition 6.1.7. — Let # be a smooth representation of G and suppose that thereexists v1, v2 & #I1 such that

(Kv1)Fp

$= "%,J , (Kv2)Fp

$= "%s,J , $"1v1 = v2, $"1v2 = v1,

then there exists a G-equivariant map ' : H0(X,V!) # # such that

'(2%,J) = v1 and '(2%s,J) = v2

where + = {(, (s}.

Proof. — By Lemma 5.4.2 and Theorem 5.5.4, it is enough to construct a morphismof diagrams D! # D(K"). However, such morphism is immediate.



Corollary 6.1.8. — Let # be a smooth representation of G and suppose that oneof the following holds: ( = (s, or p = q, then

HomG(H0(X,V!), #) $= HomH(M! , #I1 ).

Remark 6.1.9. — This fails if q != p and ( != (s. Proposition 6.4.5 gives an example.

Proof. — Lemmas 6.1.4 and 6.1.5 imply that we always have an injection

HomG(H0(X,V!), #) %"# HomH(M! , #I1).

By Lemma 2.2.2 M! |HK$= M%,J - M%s,J . Under the assumptions made, Corollaries

2.0.7, 3.1.8 and respectively 4.1.5 give us vectors v1, v2 & #I1 as in Proposition 6.1.7,hence the injection is an isomorphism.

Corollary 6.1.10. — Let # be a smooth representation, and suppose that #I1 $=M!, then

dim HomG(H0(X,V!), #) = 1.

Proof. — It is enough to consider the case p != q and ( != (s. Since Corollary 6.1.8implies the statement in the other cases. Let " = (K#I1)Fp

, then "I1 = #I1 . Hence

"I1 $= M! |HK$= M%,! - M%s,!

as an HK-module. Proposition 4.2.40 implies that " $= "%,! - "%s,!. The action of$ on #I1 is given by Corollary 2.0.7. Now we may apply Proposition 6.1.7 to geta non-zero homomorphism. So the dimension is at least one. The module M! isirreducible, and Lemmas 6.1.4 and 6.1.5 imply that the dimension is at most one.

6.2. Injective envelopes

For the convenience of the reader we recall some general facts about injectiveenvelopes. Let K be a pro-finite group and let RepK be the category of smoothFp-representations of K. We assume that K has an open normal pro-p subgroup P .

Definition 6.2.1. — Let # & RepK and let " be a K-invariant subspace of #. Wesay that # is an essential extension of " if for every non-zero K-invariant subspace #&

of #, we have #& % " != 0.Let " & RepK and let Inj be an injective object in RepK. A monomorphism

$ : " %# Inj is called an injective envelope of ", if Inj is an essential extension of $(").

Proposition 6.2.2. — Every representation " & RepK has an injective envelope$ : " %# Inj ". Moreover, injective envelopes are unique up to isomorphism.

Proof. — [16], §3.1.


6.3. ADMISSIBILITY 71

Lemma 6.2.3. — Let Inj be an injective object in RepK and let $ : " # Inj " be aninjective envelope of " in RepK. Let ' be a monomorphism ' : " %# Inj, then thereexists a monomorphism * : Inj " %# Inj such that ' = * , $.

Proof. — Since Inj is an injective object there exists * such that the diagram

0 !! " $ !!

'""

Inj "

*%%

Inj

of K-representations commutes. Since ' is an injection Ker* % $(") = 0. This impliesthat Ker* = 0, as Inj " is an essential extension of $(").

Lemma 6.2.4. — Let " & RepK be an irreducible representation and let $ : " %# Inj "be an injective envelope of " in RepK, then " %# (Inj ")P is an injective envelope of "in RepK/P .

Proof. — We note that since P is an open normal pro-p subgroup of K and " isirreducible, Lemma 2.0.2 implies that P acts trivially on ". Hence, $(") is a subspaceof (Inj ")P . Moreover, (Inj ")P is an essential extension of $("), since Inj " is anessential extension of $(").

Let L : RepK/P # RepK be a functor sending a representation - to its inflationL(-) to a representation of K, via K # K/P . Then

HomK/P(-, (Inj ")P) $= HomK(L(-), Inj ")

where the isomorphism is canonical. Since, the functor L is exact and Inj " is aninjective object in RepK , the functor HomK/P(3, (Inj ")P) is exact. Hence, (Inj ")P isan injective object in RepK/P , which establishes the Lemma.

Definition 6.2.5. — Let # & RepK, we denote by soc# the subspace of #, generatedby all irreducible subrepresentations of #.

Lemma 6.2.6. — Let " & RepK be irreducible, and let $ : " %# Inj " be an injectiveenvelope of ", then soc(Inj ") $= ".

Proof. — Let 1 be any non-zero K invariant subspace of Inj ", which is irreducible asa representation of K. Since Inj " is an essential extension of $(") and " is irreducible,we have 1 = $("). Hence, soc(Inj ") = $(").

6.3. Admissibility

Let G be a locally pro-finite group and let RepG be the category of smooth Fp-representations of G.



Definition 6.3.1. — A representation # & RepG is called admissible, if for everyopen subgroup K of G, the space #K of K-invariants is finite dimensional.

Theorem 6.3.2. — Suppose that G has an open pro-p subgroup P. A representation# & RepG is admissible if and only if #P is finite dimensional.

Proof. — If # is admissible, then #P is finite dimensional. Suppose that #P is finitedimensional and let 1 %# Inj 1 be an injective envelope of the trivial representation inRepP , then there exists *, such that the diagram

0 !! #P !!

""

#|P

*##

(dim #P) Inj 1

of P-representations commutes. This implies that (Ker*)P = 0, and hence by Lemma2.0.2, * is injective.

Let K be any open subgroup of G. Since P is an open compact subgroup of G, wemay choose an open subgroup P & of G such that P & is a subgroup of P % K and P &

is normal in P . It is enough to show that #P#is finite dimensional. Since * is an

injection, it is enough to show that (Inj1)P#

is finite dimensional. Since P is pro-pand P & is a normal open subgroup of P , Lemma 6.2.4 and Proposition 4.0.8 implythat

(Inj1)P# $= Fp[P/P &]

which is finite dimensional.

6.4. Coe!cient systems I!

Let ( : H # F!p be a character, and let

"%,J %"# Inj "%,J , "%s,J %"# Inj "%s,J

be injective envelopes of "%,J and "%s,J in RepK , respectively. We may extend theaction of K to the action of F!K, so that our fixed uniformiser !F acts trivially. Weget an exact sequence

0 "# #"%,J - #"%s,J "# 4Inj"%,J - 4Inj"%s,J

of F!K-representations. This gives a commutative diagram

0 !! #"%,J - #"%s,J!! 4Inj"%,J - 4Inj"%s,J

0 !! (#"%,J - #"%s,J)I1 !!

$$

4Inj"%,J - 4Inj"%s,J

$$


6.4. COEFFICIENT SYSTEMS I# 73

of F!I-representations. We will show that we may extend the action of F!I on(4Inj"%,J - 4Inj"%s,J)|F!I to the action of K(&1), so that we get an object Y! in DIAG,together with an embedding D! %# Y! . Since the categories DIAG and COEFG areequivalent, this will give us an embedding of coe!cient systems V! %# I! . We willshow that the image

#! = Im(H0(X,V!) "# H0(X, I!))

is an irreducible supersingular representation of G. All the hard work was done inPropositions 4.2.37 and 4.2.38, the construction of Y! and the proof of irreducibil-ity follow from the ’general non-sense’ of Section 6.2. This gives hope that similarconstruction might work for other groups.

Lemma 6.4.1. — Let " be an irreducible representation of K and let

" %"# Inj "

be an injective envelope of " in RepK , then

(Inj ")|I $=-%

dimHomH((, (inj ")U ) Inj (

where the sum is taken over all irreducible representations of H, which we identifywith the irreducible representations of I and

" %"# inj ", ( %"# Inj(

are the injective envelopes of " in Rep! and of ( in RepI , respectively.

Proof. — If ( is an irreducible representation of I, then Lemma 2.0.2 implies thatI1 acts trivially on (. Since I/I1

$= H , the irreducible representations of I and Hcoincide. Moreover, since H is abelian, all the irreducible representations of H areone dimensional. Since, the order of H is prime to p, all the representations of H aresemi-simple. Therefore

(Inj ")I1 $=-%

m%(

as a representation of I, where the multiplicity m% of ( is given by

m% = dim HomI((, Inj ").

Lemma 6.2.4 implies that (Inj ")K1 $= inj " as representations of K/K1$= ". Corollary

4.0.6 implies that inj " is finite dimensional. In particular, m% is finite for every (.Moreover,

HomI((, Inj ") $= HomI((, (Inj ")K1) $= HomB((, inj ") $= HomH((, (inj ")U ).

Hence, m% = dim HomH((, (inj ")U ). We consider an exact sequence

0 "# (Inj ")I1 "# (Inj ")|I



of I-representations. The restriction (Inj ")|I is an injective object in RepI . Lemma6.2.3 implies that

(Inj ")|I $= N --%

m% Inj(

for some representation N . Since RepH is semi-simple and Inj ( is an essential ex-tension of (, Lemma 6.2.4 implies that (Inj ()I1 $= (. By comparing the dimensionsof I1-invariants of both sides we get that dimN I1 = 0 and Lemma 2.0.2 implies thatN = 0.

Lemma 6.4.2. — Let ( : H # F!p be a character. We consider ( and (s as one

dimensional representations of I, via I/I1$= H. Let

( %"# Inj(, (s %"# Inj (s

be injective envelopes of ( and (s in RepI , respectively. Let V1 be the underlyingvector space of Inj( and let V2 be the underlying vector space of Inj(s. Further, letv1 and v2 be vectors in V1 and V2 respectively, such that

(v1)Fp= (Inj ()I1 , (v2)Fp

= (Inj(s)I1 .

Then there exists an action of K(&1) on V1 - V2, extending the action of I, so thatour fixed uniformiser !F acts trivially and

$"1v1 = v2, $"1v2 = v1.

Proof. — Let t & T be any diagonal matrix, then sts = $t$"1. Hence

(s $= ("

as I-representations, where (" denotes the action of I, on the underlying vector spaceof (, twisted by $. So we get an exact sequence

0 "# (s "# (Inj ()"

of I-representations. Twisting by $ is an exact functor in RepI and

HomI(-, (Inj ()") $= HomI(-", Inj().

Since Inj ( is an injective object in RepI , this implies that (Inj()" is an injectiveobject in RepI . Since Inj ( is an essential extension of (, (Inj()" is an essentialextension of (s. Since injective envelopes are unique up to isomorphism, there existsan isomorphism ' of I-representations

' : (Inj()" $= Inj(s.

The proof of Lemma 6.4.1 shows that the space (Inj()I1 is one dimensional. Hence,after replacing ' by a scalar multiple we may assume that '(v1) = v2. We may extendthe action of I on V1 and V2 to the action of F!I by making !F act trivially. Wedenote the corresponding representations by 4Inj( and 4Inj(s. For trivial reasons

' : (4Inj()" $= 4Inj(s.


6.4. COEFFICIENT SYSTEMS I# 75

We consider the induced representation IndK(#1)F!I

4Inj(. Let ev1 and ev" be the evalu-ation maps at 1 and $ respectively, then we get an F!I-equivariant isomorphism:

IndK(#1)F!I

4Inj( $= V1 - V2, f *"# ev1(f) + '(ev"(f)).

The action of K(&1) on the left hand side gives us the action of K(&1) on V1 -V2. Letv & V1 and w & V2, then the action of $"1 is given by

$"1(v + w) = '"1(w) + '(v)

and hence $"1v1 = v2 and $"1v2 = v1.

We will construct a diagram Y! . This will involve making some choices. Supposethat q = pn, let ( : H # F

!p be a character and let + = {(, (s}. We consider an

irreducible representation "%,J of K. Lemma 3.2.2 gives us a pair (r, a), where r isthe usual n-tuple and a is an integer modulo q"1. Let "%,J %# Inj "%,J be an injectiveenvelope of "%,J in RepK . Let Wr be the underlying vector space of Inj "%,J . Wemay assume that Wr depends only on the n-tuple r. Since, if (& = ( . (det)c, then"%#,J

$= "%,J . (det)c and a simple argument shows that "%#,J %# (Inj "%,J ). (det)c isan injective envelope of "%#,J in RepK . Let

Y!,0 = (4Inj"%,J - 4Inj"%s,J ,Wr -Wp"1"r)

where tilde denotes the extension of the action of K to the action of F!K, so that!F acts trivially. We are going to construct an action of K(&1) on Y!,0|F!I , whichextends the action of F!I, and this will give us Y! . However, this can be done in alot of ways, and not all of them suit our purposes. Lemma 6.2.4 and Remark 4.0.5imply that

(Y!,0)K1 $= inj "%,J - inj "%s,J

as K-representations, where on the right hand side we adopt the notation of Propo-sitions 4.2.37 and 4.2.38. In particular,

(Y!,0)I1 $= (inj "%,J - inj "%s,J)I1

as HK-modules. In Lemma 4.2.19 we have worked out a basis consisting of eigenvec-tors for the action of I of (a model of) (inj "%,J - inj "%s,J)I1 . The above isomorphismgives us a basis B! of (Y!,0)I1 . Lemma 6.4.1 gives an F!I-equivariant decomposition:

5 : Wr -Wp"1"r$=

-b$B#

W(b)

such that 5(b) & W(b), for every b & B! , and the representation, given by the actionof I on W(b), is an injective object in RepI , which is also an essential extension of(5(b))Fp

. To simplify things we view 5 as identification and omit it from our notation.If ( = (s then we pair up the basis vectors as in Proposition 4.2.37:

B! = {b0, b0 + b1}&

{(,1"(}###0

{b(, b1"(}.



If ( != (s then we pair up the basis vectors as in Proposition 4.2.38:

B! = {b0, b0} / {b1, b1}&

{(,1"(}###r

{b(, b1"(}&

{(,1"(}###p"1"r

{b(, b1"(}.

Let {b, b&} be any such pair and suppose that I acts on b via a character *, then Iwill act on b&, via a character *s. We denote

W(b, b&) = W(b) -W(b&).

Lemma 6.4.2 implies that there exists an action of K(&1) on W(b, b&), extending theaction of F!I, such that

$"1b = b&, $"1b& = b.

This amounts to fixing an isomorphism of vector spaces ' : W(b) $= W(b&), such that'(b) = b& and which induces an isomorphism of I representations ' : (Inj *)" $= Inj*s.

If ( = (s then Y!,0 decomposes into F!I-invariant subspaces:

W(b0, b0 + b1)-

{(,1"(}###0

W(b(, b1"().

If ( != (s then Y!,0 decomposes into F!I-invariant subspaces:

W(b0, b0) -W(b1, b1)-

{(,1"(}###r

W(b(, b1"()-

{(,1"(}###p"1"r

W(b(, b1"().

Let Y!,1 be a representation of K(&1), whose underlying vector space is Wr-Wp"1"r,and the action of K(&1) extends the action of F!I on each direct summand, as it wasdone for W(b, b&).

Definition 6.4.3. — Let Y! be an object in DIAG, given by

Y! = (Y!,0, Y!,1, id)

and let I! be the corresponding coe!cient system

I! = C(Y!).

Remark 6.4.4. — The definition of Y! depends on all the choices we have made.

Proposition 6.4.5. — Let ( : H # F!p be a character and let + = {(, (s}. Suppose

that ( = (s, thenH0(X, I!)I1 $= M!

-{(,1"(}###

0

M!$

as H-modules, where +! = +1"! = {(/!"(p"1), ((/!"(p"1))s}. Suppose that ( != (s,then

H0(X, I!)I1 $= L!-

{(,1"(}###r

M!!

-{(,1"(}###

p"1"r

M!$

as H-modules, where +! = +1"! = {(/!"(p"1"r), ((/!"(p"1"r))s} and +! = +1"! ={(s/!"r, ((s/!"r)s}.


6.5. CONSTRUCTION 77

Proof. — In Propositions 4.2.37 and 4.2.38 we have showed that we may extend theaction of HK on (inj "%,J-inj"%s,J)I1 to the action of H, so that the resulting modulesare isomorphic to the ones considered above. We will show that H0(X, I!)I1 realizesthis extension. By Proposition 5.3.5 (or alternatively Corollary 5.5.5) we have

H0(X, I!)|K(#0)$= Y!,0, H0(X, I!)|K(#1)

$= Y!,1

as K(&0) and K(&1)-representations, respectively. Moreover, the diagram

Y!,0

$= !! H0(X, I!)

Y!,1

id$$

$= !! H0(X, I!)

id$$

of F!I-representations commutes. So

(Y!,0)I1 $= H0(X, I!)I1

as HK-modules. Lemma 6.2.4 implies that

H0(X, I!)I1 $= (inj "%,J - inj "%s,J)I1

as HK-modules, and we know the right hand side from Propositions 4.2.37 and 4.2.38.It remains to determine the action of T". Corollary 2.0.7 implies that for everyv & H0(X, I!)I1 we have

vT" = $"1v.

Hence the action of T" is determined by the isomorphism

Y!,1$= H0(X, I!)|K(#1).

Since B! is a basis of (Y!,0)I1 , it is enough to know how $"1 acts on the basis vectors.Let W(b, b&) be one of the K(&1)-invariant subspaces of Y!,1, as before. We haveextended the action of F!I on Y!,0|F!I to K(&1) so that

$"1b = b&, $"1b& = b.

Hence, if we consider B! also as a basis of H0(X, I!)I1 we have

bT" = b&, b&T" = b.

Now the statement of the Proposition is just a realization of Propositions 4.2.37 and4.2.38.

6.5. Construction

Now we will construct an embedding D! %# Y! . Suppose that ( = (s, then weconsider vectors b0 and b0 + b1 in (Y!,0)I1 . Lemmas 4.2.21 and 4.2.30 imply that

(Kb0)Fp

$= #"%,S , (K(b0 + b1))Fp

$= #"%,!



as F!K-representations. We have constructed the action of K(&1) on Y!,1 so that

$"1b0 = b0 + b1, $"1(b0 + b1) = b0.

Suppose that ( != (s, then we consider vectors b0 and b0 in (Y!,0)I1 . Lemmas 4.2.21implies that

(Kb0)Fp

$= #"%,!, (Kb0)Fp

$= #"%s,!

as F!K-representations. We have constructed the action of K(&1) on Y!,1 so that

$"1b0 = b0, $"1b0 = b0.

Hence, in both cases we get an embedding D! %# Y! in the category DIAG. Thisgives us an embedding of G equivariant coe!cient systems V! %# I! .

Definition 6.5.1. — Let #! be a representation of G, given by

#! = Im(H0(X,V!) "# H0(X, I!)).

Theorem 6.5.2. — For each + = {(, (s}, the representation #! is irreducible andsupersingular. Moreover, #I1

! contains an H-submodule isomorphic to M!. Further, if

#!$= #!#

then + = +&.

Proof. — Lemma 5.3.2 implies that #! is non-zero. So by Corollary 6.1.6 it is enoughto prove that #! is irreducible. To ease the notation we identify the underlying vectorspaces of Y!,0 and H0(X, I!). If ( = (s then Lemma 6.1.5 implies that

#! = (Gb0)Fp= (G(b0 + b1))Fp

.

If ( != (s then Lemma 6.1.5 implies that

#! = (Gb0)Fp= (Gb0)Fp

.

This can be rephrased in a di#erent way. By Proposition 5.3.5 we have

H0(X, I!)|K $= Inj "%,J - Inj "%s,J

as K-representations. Lemma 6.2.6 implies that

"%,J - "%s,J$= soc(H0(X, I!)|K).

Hence, if ( = (s then

(soc(H0(X, I!)|K))I1 = (b0, b0 + b1)Fp

and if ( != (s then(soc(H0(X, I!)|K))I1 = (b0, b0)Fp

and hence#! = (G(soc(H0(X, I!)|K))I1 )Fp

.



The key point is that (soc(H0(X, I!)|K))I1 is an H-invariant subspace of H0(X, I!)I1 ,moreover

(soc(H0(X, I!)|K))I1 $= M!

as an H-module. This can be deduced either from Lemma 6.1.4 or from the modulecomputation in Proposition 6.4.5.

Suppose that #& is non-zero G-invariant subspace of #! then by Lemma 2.0.2(#&)K1 != 0, and hence soc(#&|K) != 0. We apply Lemma 2.0.2 again to obtain(soc(#&|K))I1 != 0. We have trivially soc(#&|K) + soc(H0(X, I!)|K). Hence

0 != (soc(#&|K))I1 # (soc(H0(X, I!)|K))I1 % (#&)I1 .

Since the spaces (soc(H0(X, I!)|K))I1 and (#&)I1 are H-invariant subspaces ofH0(X, I!)I1 , and (soc(H0(X, I!)|K))I1 is an irreducible H-module, we get

(soc(H0(X, I!)|K))I1 # (#&)I1

and this implies that #& = #! . Hence #! is irreducible.Suppose that #!

$= #!# , then this induces an isomorphism of vector spaces

' : (soc(#! |K))I1 $= (soc(#!# |K))I1 .

The argument above implies that both spaces are H-invariant and Corollary 2.0.7implies that ' is an isomorphism of H-modules. Hence,

M!$= (soc(#! |K))I1 $= (soc(#!# |K))I1 $= M!# .

Lemma 2.1.3 implies that + = +&.

Corollary 6.5.3. — The representation H0(X, I!) is an essential extension of #!

in RepG. In particular,#!

$= soc(H0(X, I!)),where soc(H0(X, I!)) is the subspace of H0(X, I!) generated by all the irreduciblesubrepresentations.

Proof. — Let # be a non-zero G-invariant subspace of H0(X, I!). The proof of The-orem 6.5.2 shows that (soc(H0(X, I!)|K))I1 is a subspace of #I1 . This implies that#! is a subspace of #. The last part is immediate.

6.5.1. Twists by unramified quasi-characters. — Let , & F!p , we define an

unramified quasi-character µ& : F! # F!p , by

µ&(x) = ,valF (x).

Lemma 6.5.4. — Suppose that #! . µ& , det $= #!# , then + = +& and , = ±1.

Proof. — Our fixed uniformiser !F acts on #! . µ& , det, by a scalar ,2, and it actstrivially on #!# . Hence, , = ±1. By Lemma 2.1.9 M! . µ"1 , det $= M! , and henceby the argument of 6.5.2 M!# $= M! , which implies that + = +&.



Proposition 6.5.5. — Suppose that q = p, then #! . (µ"1 , det) $= #! .

Proof. — By Corollary 6.5.3 it is enough to show that Y!.(µ"1,det) $= Y! in DIAG.We claim that we always have

Y!,1$= Y!,1 . (µ"1 , det)

as K(&1)-representations. Since F!I is contained in the kernel of µ"1 , det, it isenough to examine the action of $. We recall that the action of K(&1) was defined,by fixing a certain isomorphism ' : W(b) $= W(b&), and then letting $"1 act onW(b, b&) = W(b) -W(b&) by

$"1(v + w) = '"1(w) + '(v).

Let $1 be an F!I-equivariant isomorphism

$1 : W(b) -W(b&) $= W(b) -W(b&), v + w *"# v " w,

then, since µ"1(det($"1)) = "1, we have

$"1 . µ"1(det($"1))($1(v + w)) = '"1(w) " '(v) = $1($"1(v + w)).

Hence W(b, b&) $= W(b, b&) . (µ"1 , det) as K(&1)-representations and hence Y!,1$=

Y!,1 . (µ"1 , det) as K(&1)-representations. Since F!K is contained in the kernel ofµ"1 ,det we also have Y!,0

$= Y!,0. (µ"1 ,det). However, to define an isomorphism inDIAG we need to find $0 : Y!,0

$= Y!,0, which is compatible with $1 via the restrictionmaps. If p = q this is easy, since if ( = (s, then

Wr -Wp"1"r = W(b0) -W(b0 + b1)

and if ( != (s then

Wr -Wp"1"r = (W(b0) -W(b1)) - (W(b0) -W(b1))

and the subspaces that $ ‘swaps’ come from di#erent injective envelopes. Note, thatthis is not the case if q != p. Hence, if we define

$0 : Wr -Wp"1"r$= Wr -Wp"1"r, v + w *"# v " w

then $ = ($0, $1) is an isomorphism $ : Y!$= Y! . (µ"1 , det).

Lemma 6.5.6. — The representations H0(X, I!) and #! are admissible.

Proof. — Proposition 6.4.5, Lemma 6.3.2.

Our main result can be summarised as follows.

Theorem 6.5.7. — Let !F be a fixed uniformiser, then there exists at least q(q"1)/2pairwise non-isomorphic, irreducible, supersingular, admissible representations of G,which admit a central character, such that !F acts trivially.



Proof. — There are precisely q(q " 1)/2 orbits + = {(, (s}. Then the statementfollows from Theorem 6.5.2 and Corollary 6.5.6. Each #! admits a central character,since H0(X,V!) admits a central character. If , & o!F , then it acts on H0(X,V!) bya scalar

(++

& 00 &

,,= (s

++& 00 &

,,

and !F acts trivially by construction.

If F = Qp then we may apply the results of Breuil [4].

Corollary 6.5.8. — Suppose that F = Qp, then #! is independent up to isomor-phism of the choices made in the construction of Y! . Moreover, if # is an irreduciblesupersingular representation of G, admitting a central character, then there exists, & F

!p , unique up to a sign, and a unique +, such that

# $= #! . (µ& , det).

Proof. — In [4] Breuil has determined all the supersingular representations, in thecase of F = Qp. As a consequence, by [18] Theorem E.7.2, the functor of I1-invariants,RepG # Mod -H, # *# #I1 induces a bijection between the isomorphism classes ofirreducible supersingular representations with a central character and isomorphismclasses of supersingular right modules of H. In particular, there are precisely p(p"1)/2isomorphism classes of supersingular representations with a central character, suchthat !F acts trivially. By Theorem 6.5.7 our construction yields at least p(p " 1)/2such representations. Hence #! does not depend up to isomorphism on the choicesmade for Y! .

Let # be any supersingular representation of G with a central character. We mayalways twist # by an unramified quasi-character, so that !F acts trivially. Hence byabove

# $= #! . (µ& , det)and by Lemma 6.5.4 and Proposition 6.5.5, + is determined uniquely and , up to±1.


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