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Geometric Realizations of the Basic Representation of the

Affine General Linear Lie Algebra

Joel Lemay

Thesis submitted to the Faculty of Graduate and Postdoctoral Studies in partial

fulfillment of the requirements for the degree of

Doctor of Philosophy in Mathematics 1

Department of Mathematics and Statistics

Faculty of Science

University of Ottawa

© Joel Lemay, Ottawa, Canada, 2015

1The Ph.D. program is a joint program with Carleton University, administered by the Ottawa-Carleton Institute of Mathematics and Statistics

Abstract

The realizations of the basic representation of glr are well-known to be parametrized

by partitions of r and have an explicit description in terms of vertex operators on

the bosonic/fermionic Fock space. In this thesis, we give a geometric interpreta-

tion of these realizations in terms of geometric operators acting on the equivariant

cohomology of certain Nakajima quiver varieties.

ii

Resume

Il est bien connu que les realizations de la representation de base de glr sont parametrises

par les partitions de r et que chacune de ces realization possede une description ex-

plicite en termes d’operateurs de sommet qui agissent sur l’espace de Fock bosonique

et fermionique. Dans cette these, nous donnons une interpretation geometrique de

ces realizations en termes d’operateurs geometriques qui agissent sur la cohomologie

equivariante des varietees de carquois de Nakajima.

iii

Acknowledgements

I would like to thank my mother and father for always being there for me and sup-

porting me in all my endeavours. All my successes in life begin with you.

Moreover, I would like to thank Alistair Savage not only for his invaluable help

throughout the writing of this thesis, but also for his guidance and encouragement

throughout my graduate studies. I would also like to thank Yuly Billig for his helpful

advice.

Finally, I thank the University of Ottawa, NSERC and OGS for their financial

support during my Ph.D. studies.

iv

Contents

1 The Basic Representation of the Affine General Linear Lie Alge-

bra 4

2 Geometric Invariant Theory 28

3 Quiver Varieties 42

4 Vector Bundles and Geometric Operators 63

5 Geometric Realizations of the Basic Representation 83

v

Introduction

Let g be a semisimple Lie algebra and denote the corresponding untwisted affine Lie

algebra by g. The basic representation of g, which we denote by Vbasic = Vbasic(g), is

the irreducible highest weight representation whose highest weight is the fundamental

weight corresponding to the additional node of the affine Dynkin diagram (compared

to the corresponding finite Dynkin diagram). The basic representation is so-named

since it is, in a sense, the simplest nontrivial representation of g. In the late 70’s and

early 80’s mathematicians began constructing the first explicit realizations of Vbasic.

The first such realization was given by Lepowsky and Wilson in [14] for Vbasic(sl2).

Their construction was later generalized to arbitrary simply-laced affine Lie algebras

and twisted affine Lie algebras in [10], and this construction became known as the

principal realization of Vbasic. However, Frenkel and Kac in [6], and Segal in [25],

gave an entirely different realization of Vbasic; this construction was referred to as the

homogeneous realization. While the principal and homogeneous realizations seemed

completely unrelated, it was discovered by Kac and Peterson in [12], and by Lepowsky

in [13], that the two realizations depend implicitly on the choice of a so-called maximal

Heisenberg subalgebra of g. Indeed, one can associate a realization of Vbasic to each

maximal Heisenberg subalgebra of g.

In this thesis, we will focus on the case where g = glr. While glr is not semisimple,

it is a one-dimensional central extension of the semisimple Lie algebra slr, and thus has

a similar representation theory. Up to conjugacy under the adjoint action of the Kac-

1

CONTENTS 2

Moody group, the maximal Heisenberg subalgebras of an affine Lie algebra are known

to be parametrized by the conjugacy classes of the Weyl group of the corresponding

finite-dimensional Lie algebra (see [12, Proposition of Section 9]). The Weyl group of

glr is the symmetric group on r elements, Sr, and the conjugacy classes of Sr are in

one-to-one correspondence with partitions of r, i.e. s-tuples (r1, . . . , rs) ∈ (N+)s such

that

r = r1 + · · ·+ rs, and r1 ≤ · · · ≤ rs.

Thus, there exists a realization of Vbasic(glr) for each partition of r, the principal

and homogeneous realizations corresponding to the two extreme partitions (r) and

(1, . . . , 1), respectively. The realizations for every partition of r were described by

ten Kroode and van de Leur in [26] using vertex operators acting on bosonic Fock

space (a representation of the Heisenberg algebra) and fermionic Fock space (a repre-

sentation of the Clifford algebra). More precisely, for each partition of r, there exists

a precise vector space isomorphism between bosonic Fock space and fermionic Fock

space (known as the boson-fermion correspondence), and thus the Heisenberg and

Clifford algebras may be thought of as operators acting on a common space. The

construction in [26] defines a representation of glr on bosonic/fermionic Fock space in

terms of vertex operators (i.e. formal power series) of Heisenberg and Clifford algebra

operators. The so-called “zero-charge” subspace of bosonic/fermionic Fock space is

then shown to be isomorphic, as a glr-representation, to Vbasic.

In this thesis, we give a geometric interpretation of these algebraic realizations

of Vbasic(glr). Our general strategy is as follows. We fix a partition (r1, . . . , rs) of r

and consider the moduli space of framed torsion-free sheaves of rank r and second

Chern class n, M(s, n). In [15], Licata and Savage showed that, under a suitable

torus action, the localized equivariant cohomology of (a disjoint union of infinitely-

many copies) ofM(s, n) provides a suitable geometric analogue of bosonic/fermionic

Fock space. This is accomplished by defining an action of the Heisenberg and Clifford

CONTENTS 3

algebras on this cohomology in terms of the top Chern classes of certain equivariant

vector bundles on M(s, n). The construction given in [15] naturally corresponds to

the homogeneous realization in [26], and thus we generalize their construction to an

arbitrary partition. With this framework in place, we define a new set of operators

using vector bundles on certain subvarieties of M(s, n). We then show that these

operators may be expressed as vertex operators of our “geometric” Heisenberg and

Clifford algebra operators, and that the formulas we obtain exactly match those

found in the algebraic realizations of the representation of glr on Vbasic, thus giving

us geometric realizations of Vbasic.

The paper is organized into 5 chapters. In Chapter 1, we review the Heisenberg

and Clifford algebra representations on bosonic and fermionic Fock space. We also

briefly summarize the various algebraic realizations of Vbasic found in [26]. In Chapter

2, we recall some of the basic concepts from algebraic geometry, especially geometric

invariant theory, that we will use in subsequent chapters. In Chapter 3, we will

discuss our main geometric object of interest: Nakajima quiver varieties (of which the

aforementioned moduli space M(s, n) is a special case). Chapter 4 will describe our

method of constructing geometric operators on the localized equivariant cohomology

of quiver varieties from equivariant vector bundles. Finally, in Chapter 5, we define

our geometric analogues of the action of the Heisenberg algebra, Clifford algebra, and

glr on bosonic and fermionic Fock space. We also present our main theorem (Theorem

5.21), which is a geometric analogue of Theorem 1.20 (the main theorem of [26]).

Chapter 1

The Basic Representation of the

Affine General Linear Lie Algebra

In this first chapter, we will summarize the known algebraic realizations of the basic

representation of glr. In particular, the inequivalent realizations are parametrized by

the different partitions of r. For a more in-depth treatment of this topic, the reader

is encouraged to see [26] or [11]. The goal in the subsequent chapters will be to give

a geometric construction of the representations presented here.

We begin with a description of the s-coloured oscillator algebra and s-coloured

Clifford algebra, along with the associated s-coloured bosonic and fermionic Fock

spaces.

Definition 1.1. (s-coloured oscillator algebra) For s ∈ N+, the s-coloured oscillator

algebra, s, is the complex Lie algebra

s :=s⊕`=1

(⊕n∈Z

CP`(n)

)⊕ Cc,

with the Lie bracket determined by

[s, P`(0)] = 0, [P`(n), Pk(m)] =1

nδ`,kδn+m,0c, n 6= 0,

for all `, k = 1, . . . , s and m,n ∈ Z.

4

1. The Basic Representation of the Affine General Linear Lie Algebra 5

Remark 1.2. Our presentation of the s-coloured oscillator algebra differs slightly

from the one sometimes found elsewhere in the literature. In particular, in [26,

Section 1], s is defined as the complex Lie algebra with basis α`(n)n∈Z,`=1,...,s ∪ c

and Lie bracket given by

[α`(n), αk(m)] = nδ`,kδn+m,0c.

The connection between the two presentations is made by setting

P`(n) =1

|n|α`(n), n 6= 0, and P`(0) = α`(0).

We favour the presentation given in Definition 1.1 since this one turns out to be more

natural from a geometric point of view.

Definition 1.3. (s-coloured Heisenberg algebra) The subalgebra

s0 =s⊕`=1

⊕n∈Z−0

CP`(n)

⊕ Cc,

of s is the so-called s-coloured Heisenberg algebra and, as we will see later, it serves

as the main ingredient in the various realizations of the basic representation of glr.

Let Λ ⊆ C[t1, t2, . . . ] denote the ring of symmetric functions in infinitely many

variables. It is well-known that

Λ = C[p1, p2, . . . ],

where pn is the n-th power sum

pn =∞∑i=1

tni .

We define the s-coloured bosonic Fock space to be the space

B := B⊗s, where B := Λ⊗C C[q, q−1].


We define a Z-grading on B given by

B =⊕c∈Z

Bc, Bc := Λ⊗ Cqc.

This induces a Zs-grading on B given by

B =⊕c∈Zs

Bc, Bc := Bc1 ⊗ · · · ⊗Bcs .

For c = (c1, . . . , cs) ∈ Zs, we use the notation |c| = c1 + · · · + cs. We then have a

Z-grading on B given by

B =⊕c∈Z

B(c), B(c) =⊕|c|=c

Bc.

One can easily verify that the mapping

P`(n) 7→ 1⊗`−1 ⊗ ∂

∂pn⊗ 1⊗s−`, n > 0,

P`(−n) 7→ 1⊗`−1 ⊗ 1

npn ⊗ 1⊗s−`, n > 0,

P`(0) 7→ 1⊗`−1 ⊗ q ∂∂q⊗ 1⊗s−`, c 7→ 1,

defines an irreducible representation of s on B.

Remark 1.4. Again, our definition of s-coloured bosonic Fock space differs slightly

from the definition sometimes found elsewhere in the literature. In particular, s-

coloured bosonic Fock space is often only considered as a representation of the s-

coloured Heisenberg algebra (rather than of the full oscillator algebra). The s-coloured

Heisenberg subalgebra acts trivially on the C[q, q−1] factors of bosonic Fock space,

and so, as an s0-module, B is isomorphic to a direct sum of infintely many copies of

Λ⊗s. In [26, Section 1], bosonic Fock space is defined as

C[x1, x2, . . . ]⊗s,


with the action of the Heisenberg algebra given by

α`(n) 7→ 1⊗`−1 ⊗ ∂

∂xn⊗ 1⊗s−`, n > 0,

α`(−n) 7→ 1⊗`−1 ⊗ nxn ⊗ 1⊗s−`, n > 0,

c 7→ 1.

However, one has an isomorphism (a priori, only of rings) Λ → C[x1, x2, . . . ], de-

termined by the mapping pn 7→ nxn. This in turn induces an isomorphism Λ⊗s →

C[x1, x2, . . . ]⊗s. One easily checks that the diagram

Λ⊗s C[x1, x2, . . . ]⊗s

Λ⊗s C[x1, x2, . . . ]⊗s

|n|P`(n) α`(n)

commutes for all ` = 1, . . . , s and n ∈ Z − 0 (this justifies the statement from

Remark 1.2 that P`(n) = 1|n|α`(n)). Thus, the map Λ⊗s → C[x1, x2, . . . ]

⊗s is in fact

an isomorphism of s-coloured Heisenberg algebra representations.

Remark 1.5. Notice that the sets 1, . . . , s × N+ and N+ are countable. Pick a

bijection ϕ : 1, . . . , s × N+ → N+. Then the mapping

P`(k) 7→ P (ϕ(`, k)), P`(−k) 7→ ϕ(`, k)

kP (−ϕ(`, k)), c 7→ c,

for all ` ∈ 1, . . . , s and k ∈ N+, is an isomorphism from the s-coloured Heisen-

berg algebra to the 1-coloured Heisenberg algebra. Thus, we will often use the term

“Heisenberg algebra” without specifying the number of colours. Moreover, one can

show (see, for instance, [11, Lemma 14.4(a)]) that Λ⊗s and Λ are isomorphic as Heisen-

berg algebra modules. The reason for the s-coloured presentation of the Heisenberg

algebra will become clear when we describe the various realizations of the basic rep-

resentation of glr.


Definition 1.6. (s-coloured Clifford algebra) The s-coloured Clifford algebra, Cl, is

the complex associative algebra generated by elements ψ`(i), ψ∗` (i), ` = 1, . . . , s, i ∈ Z,

and relations

ψ`(i), ψ∗k(j) = δijδ`k, ψ`(i), ψk(j) = ψ∗` (i), ψ∗k(j) = 0.

where x, y = xy + yx.

A semi-infinite monomial is an expression of the form

I = i1 ∧ i2 ∧ i3 ∧ · · · ,

where the in are integers such that

i1 > i2 > i3 > . . . , in+1 = in − 1, for n 0.

The charge of a semi-infinite monomial I is the integer c = c(I) such that

in = c− n+ 1, for all n 0.

The energy of a semi-infinite monomial is∑n∈N+

in − (c− n+ 1).

Let F be the complex vector space with basis the set of all semi-infinite monomials.

The charge induces a grading on F :

F =⊕c∈Z

Fc,

where Fc is the subspace with basis the set of semi-infinite monomials of charge c.

Define wedging and contracting operators ψ(i) and ψ∗(i), i ∈ Z, on F by

ψ(i)(i1 ∧ i2 ∧ · · · ) =

(−1)k(i1 ∧ · · · ∧ ik ∧ i ∧ ik+1 ∧ · · · ), if ik > i > ik+1,

0, if i = in, for some n,


ψ∗(i)(i1 ∧ i2 ∧ · · · ) =

(−1)k−1(i1 ∧ · · · ∧ ik−1 ∧ ik+1 ∧ · · · ), if i = ik,

0, if i 6= in, for all n.

It is easy to see that, when they do not act as zero, the wedging and contracting

operators raise and lower the charge of a semi-infinite monomial by 1, and so

ψ(i) : Fc → Fc+1, ψ∗(i) : Fc → Fc−1,

for all i, c ∈ Z.

We define s-coloured fermionic Fock space to be

F := F⊗s.

By our decomposition of F in terms of the charge, we have a Zs-grading

F =⊕c∈Zs

Fc, Fc := Fc1 ⊗ · · · ⊗ Fcs .

This induces a Z-grading on F given by

F =⊕c∈Z

F(c), F(c) :=⊕|c|=c

Fc.

One can show that we have a representation of Cl on F given by:

ψ`(i)|Fc 7→ (−1)c1+···+c`−1(1⊗`−1 ⊗ ψ(i)⊗ 1⊗s−`

),

ψ∗` (i)|Fc 7→ (−1)c1+···+c`−1(1⊗`−1 ⊗ ψ∗(i)⊗ 1⊗s−`

).

One can also show that F is a faithful, irreducible representation of Cl. In fact, F is

generated by the so-called vacuum vector, |0〉⊗s, where

|0〉 = 0 ∧ −1 ∧ −2 ∧ · · · .

Note that F is referred to as the spin module in [26].


Remark 1.7. We have used here the convention that Clifford algebra generators of

different colours anti-commute. However, the Clifford algebra is sometimes defined

by letting generators of different colours commute (see for example [15, Section 1.2]).

In this case it is necessary to modify the action of Cl on F to

ψ`(i)|Fc 7→ 1⊗`−1 ⊗ ψ(i)⊗ 1⊗s−`,

ψ∗` (i)|Fc 7→ 1⊗`−1 ⊗ ψ∗(i)⊗ 1⊗s−`.

Remark 1.8. By an argument completely analogous to Remark 1.5, the s-coloured

Clifford algebra is isomorphic to the 1-coloured algebra, and, by identifying the two

algebras, s-coloured fermionic Fock space is isomorphic to 1-coloured fermionic Fock

space. Thus, we will often refer to the Clifford algebra and fermionic Fock space

without specifying the number of colours.

In the sequel, it will be useful to think of fermionic Fock space not only in terms

of semi-infinite monomials, but also in terms of Young diagrams. Recall that a Young

diagram is a finite collection of boxes arranged in rows and columns such that the

number of boxes in the i-th row is greater than or equal to the number of boxes in

the (i+ 1)-st row. There are different conventions regarding the way to draw Young

diagrams, but we will choose the English notation. That is, our rows will be left-

justified and subsequent rows are placed underneath the previous one, as illustrated

in Figure 1.1. We say that a box is in the (i, j)-th position if it is in the i-th row and

j-th column of the diagram. For example, in Figure 1.1, the box labelled with a “?”

is in the (3, 2)-th position. We may thus view a Young diagram λ as being a subset

of (N+)2, where (i, j) ∈ λ if and only if λ has a box in the (i, j)-th position. For any

Young diagram λ and any k ∈ N+, let λk denote the number of boxes in its k-th row

(λk = 0 if there are no boxes in the k-th row). For any (i, j) ∈ (N+)2, define the arm

and leg length of (i, j) to be

aλ(i, j) := λi − j and `λ(i, j) := maxi′ ∈ N+ | (i′, j) ∈ λ − i,


?

Figure 1.1: Young diagram

respectively. Intuitively, for each (i, j) ∈ λ, aλ(i, j) and `λ(i, j) count the number of

boxes to the right of and below (i, j), respectively. For two Young diagrams λ and µ,

we define the relative hook length of (i, j) to be

hλ,µ(i, j) := aλ(i, j) + `µ(i, j) + 1. (1.1)

If λ = µ, we will simply write hλ(i, j) := hλ,λ(i, j). Finally, we define the residue of

box in the (i, j)-th position to be j − i.

Clearly, every weakly decreasing sequence of non-negative integers

(λ1, λ2, λ3, . . . ),

such that λk = 0 for k 0 uniquely defines a Young diagram. Thus, one has a

bijection between the set of semi-infinite monomials of a fixed charge and the set of

Young diagrams. Indeed, suppose

I = i1 ∧ i2 ∧ i3 ∧ · · · ,

is a semi-infinite monomial of charge c. Define λ(I) to be the Young diagram deter-

mined by

λ(I)k = ik − c+ k − 1. (1.2)

Lemma 1.9. The mapping I 7→ λ(I) is a bijection from the set of semi-infinite

monomials of charge c to the set of Young diagrams. Moreover, if I is a semi-infinite

monomial of energy n, then λ(I) consists of n boxes.


Hence, by Lemma 1.9, we may think of Fc as the complex vector space with

basis the set of all Young diagrams, thus giving us a Young diagram interpretation of

fermionic Fock space.

The description of the realizations of the basic representation of glr will rely

on the precise relationship between bosonic and fermionic Fock spaces known as the

boson-fermion correspondence. It is well-known that the set of Schur functions sλ,

where λ runs over all Young diagrams, is a basis of Λ (see statement (3.3) of [17]).

We then have a vector space isomorphism

Bc → Fc,

sλ ⊗ qc 7→ λ,(1.3)

for each c ∈ Z. This induces isomorphisms B(c)∼=−→ F(c) and B

∼=−→ F.

We turn our attention now to the representation theory of Lie algebras. Let g

be a matrix Lie algebra and let g = g⊗C[t, t−1] be the loop algebra on g. The affine

Lie algebra of g is

g := g⊕ Cc,

with Lie bracket given by

[g, c] = 0, [x⊗ tn, y ⊗ tm] = [x, y]⊗ tn+m + nδn+m,0 tr(xy)c. (1.4)

Here we follow the conventions in [26, Equation (4.2.5)] (note that the two-cocycle

µ in [26] simplifies to a multiple of the trace map when restricted to elements of the

form x⊗ tn).

Remark 1.10. In the literature, the Lie bracket on g is usually defined as

[x⊗ tn, y ⊗ tm] = [x, y]⊗ tn+m + nδn+m,0〈x|y〉c,

where 〈x|y〉 = tr(adx ad y) is the Killing form on g. For each of the classical Lie

algebras, the Killing form is just a (nonzero) multiple of the trace map. In particular,


for slr,

〈x|y〉 = 2r tr(xy).

Thus, up to a scaling factor on c, there is no difference between using the Killing form

and the trace map. Meanwhile, on glr, the Killing form is degenerate since

〈I|x〉 = 0,

for all x ∈ glr, whereas the trace map is nondegenerate, making it a more appropriate

choice. This is our motivation for defining the Lie bracket on g by (1.4).

Remark 1.11. The loop algebra g is sometimes defined as

g =⊕n∈Z

einθg,

where θ is a formal parameter. Of course, one has the obvious isomorphism of Lie

algebras ⊕n∈Z

einθg→ g⊗ C[t, t−1],

einθx 7→ x⊗ tn,

and so both definitions are equivalent. We will use this alternative description when

it is more convenient.

Next, we describe the basic representation, Vbasic = Vbasic(g), of g when g = slr

or glr, r ∈ N+ (although the description applies to other Lie algebras as well). Let

Ei,j1≤i,j≤r be the standard basis of glr. That is Ei,j is the r×r matrix with (i, j)-th

entry 1 and zeros elsewhere. Recall that, for r ≥ 2, slr is generated by its Chevalley

generators Ei, Fi, Hi, i = 0, 1, . . . , r − 1, which are given by

Ei = Ei,i+1 ⊗ 1, Fi = Ei+1,i ⊗ 1, Hi = (Ei,i − Ei+1,i+1)⊗ 1, i = 1, . . . , r − 1,

E0 = Er,1 ⊗ t, F0 = E1,r ⊗ t−1, H0 = ((Er,r − E1,1)⊗ 1) + c.

The Chevalley generators of slr satisfy the well-known Kac-Moody relations (for slr):


1. [Hi, Hj] = 0,

2. [Ei, Fj] = δijHi,

3. [Hi, Ej] = aijEj,

4. [Hi, Fj] = −aijFj,

5. (adEi)(1−aij)(Ej) = 0, for i 6= j,

6. (adFi)(1−aij)(Fj) = 0, for i 6= j,

where

aij =

2, if i = j,

−2, if i 6= j,

for r = 2,

aij =

2, if i = j,

−1, if |i− j| = 1,

0, otherwise,

for r > 2.

Note that the r× r matrix (aij) is called the generalized Cartan matrix associated to

slr. We define the triangular decomposition of slr:

slr = sl+

r ⊕ h⊕ sl−r ,

where sl+

r (resp. sl−r ) is the subalgebra generated by the Ei (resp. the Fi) and h is the

abelian subalgebra with basis Hi. The subalgebra h is called a Cartan subalgebra

of slr. Given γ = (γ0, γ1, . . . , γr−1) ∈ Zr, a highest weight representation, V , of slr is

a representation such that there exists a vector v0 ∈ V satisfying

Ei · v0 = 0, and Hi · v0 = γiv0, (1.5)

for all i = 0, 1, . . . , r − 1, and

U(slr) · v0 = V,


where U(slr) is the universal enveloping algebra of slr. Note that, in terms of the

triangular decomposition of slr, (1.5) is equivalent to

sl+

r · v0 = 0, and Hi · v0 = γiv0.

The r-tuple γ is called the highest weight of V and the vector v0 is called a highest

weight vector of V . If V is irreducible, one can show that any two highest weight

vectors are proportional, thus, in this case, we will often abuse terminology and call

v0 the highest weight vector of V . Given a highest weight representation V of highest

weight γ, for each β ∈ Zr, define

Vβ :=

v ∈ V

∣∣∣∣∣ Hi · v =

(γi −

r−1∑j=0

aijβj

)v

.

Clearly, v0 ∈ V0. Let 1i denote the r-tuple with 1 in its i-th entry and zeros elsewhere.

It follows from the Kac-Moody relations that, restricted to Vβ,

Ei, Fi : Vβ → Vβ∓1i ,

and so

V =⊕β∈Zr

Vβ. (1.6)

We call the decomposition in (1.6) the weight space decomposition of V .

Remark 1.12. In our treatment of weights above, we are slightly abusing terminol-

ogy. Given a semisimple Lie algebra g and a Cartan subalgebra h ⊆ g, a weight is

defined to be a linear map

β : h→ C,

and the associated weight space decomposition of a representation V is

V =⊕β

Vβ, Vβ = v ∈ V | h · v = β(h)v for all h ∈ h.

However, the Chevalley generators H0, H1, · · ·Hr−1 form a basis of our Cartan sub-

algebra h and thus induce a basis of the vector space of weights. Our description

merely identifies the weight with its coefficients with respect to this basis.


Definition 1.13 (Basic representation of slr). The basic representation of slr, which

we denote Vbasic(slr), is the (unique) irreducible highest weight representation with

highest weight γ = (1, 0, . . . , 0).

Remark 1.14. See [11, Proposition 9.3] for a proof that there is a unique irreducible

highest weight representation of slr with highest weight γ = (1, 0, . . . , 0).

There is another way to describe Vbasic(slr) that amounts to the same thing.

Namely, one can define Vbasic(slr) to be the unique irreducible representation that

admits a vector v0 satisfying

(slr ⊗ C[t]) · v0 = 0, and c · v0 = v0.

It is easy to check that this alternate description is equivalent to Definition 1.13

and that v0 here plays the role of the highest weight vector. The advantage to this

construction is that it naturally generalizes to glr.

Definition 1.15 (Basic representation of glr). The basic representation of glr, which

we denote Vbasic(glr), is the (unique) irreducible representation that admits a vector

v0 (highest weight vector) such that

(glr ⊗ C[t]) · v0 = 0, and c · v0 = v0.

From now on, unless otherwise specified, Vbasic will always mean Vbasic(glr).

Remark 1.16. The proof that Vbasic is unique is essentially the same as [11, Propo-

sition 9.3].

Remark 1.17. There is a way to describe Vbasic that is more akin to Definition 1.13

by introducing a triangular decomposition of glr. Firstly, we observe that

glr = slr ⊕ CI,

and so

glr = slr ⊕(I ⊗ C[t, t−1]

).


Thus, we can define a triangular decomposition of glr by extending the triangular

decomposition of slr. Namely, we write

glr = gl+

r ⊕ h⊕ gl−r ,

where

gl±r := sl

±r ⊕

(⊕k∈N+

C(I ⊗ t±k)

),

and h is the Cartan subalgebra with basis H0, H1, . . . , Hr−1, (I ⊗ 1). We can then

define Vbasic to be the unique irreducible representation that admits a vector v0 (high-

est weight vector) satisfying

gl+

r · v0 = 0, Hi · v0 = δi,0v0, (I ⊗ 1) · v0 = 0.

Our next task is to describe the various vertex operator realizations of Vbasic as

given in [26]. The motivation for these different realizations comes from the following

proposition.

Proposition 1.18. Let V be a representation of the 1-coloured Heisenberg algebra

such that c acts as the identity and there exists an N ∈ N such that

α(n1) · · ·α(nk) · v = 0,

for all v ∈ V and ni ∈ N+ such that n1 + · · · + nk > N . Then V is isomorphic to a

direct sum of copies of Λ.

Proof: This is [11, Lemma 14.4(b)] (where the fact that c acts as the identity

implies a = 1).

Suppose that g is an affine Lie algebra with underlying finite dimensional Lie

algebra g and V is a representation of g. Furthermore, suppose that we have a (1-

coloured) Heisenberg subalgebra s0 ⊆ g such that the restriction of V to s0 satisfies


the hypotheses of Proposition 1.18. Then, by Proposition 1.18, one can, in principle,

give a realization of V completely in terms of (a direct sum of copies of) Λ. That is,

V ∼=⊕i∈I

Λ,

as s0-modules, for some indexing set I. Define the vacuum space, Ω = Ω(V ) to be

the subspace

Ω :=v ∈ V | α(k) · v = 0 for all k ∈ N+

.

It is an easy exercise to see that 1ii∈I is a basis of Ω(⊕

i∈I Λ). Therefore,

V ∼= Ω(V )⊗ Λ,

as s0-modules, where the action of s0 on Ω(V )⊗ Λ is given by

α(k) · (v ⊗ f) = v ⊗ α(k) · f,

for all k ∈ Z − 0 and v ⊗ f ∈ Ω(V ) ⊗ Λ. Of course, in general, one has many

Heisenberg subalgebras of g, and so we naturally have several realizations of V . When

g is semisimple, a complete classification of the Heisenberg subalgebras of g is given

in [12, Section 9]. We briefly summarize the classification below.

Let W be the Weyl group of g. Fix a Cartan subalgebra h ⊆ g and an element

w ∈ W . Let m be the order of w. Recall that there is an action of W on h induced

by the action of W on the root space of g. Write

h =m−1⊕k=0

hk, hk := h ∈ h | w · h = e−2πik/mh.

There exists an x ∈ g such that Ad(exp 2πix)|h = w and [x, h0] = 0 (see, for instance,

[8, Section 14.3] and [4, Theorem 2.5.5]). For each h = (h0, h1, . . . , hm−1) ∈ h and

k ∈ Z, define h(k) to be the loop

h(k) = eiθk/m Ad(exp iθx)hk ∈ g, (1.7)


where k is the unique integer 0 ≤ k ≤ m− 1 such that k ≡ k mod m. Then

sw0 :=

(⊕k∈Z

h(k)

)⊕ Cc, (1.8)

is a maximal Heisenberg subalgebra of g. Let W ′ be a set of representatives of

the conjugacy classes of W . Then the subalgebras sw0 , w ∈ W ′, form a complete

nonredundant list of maximal Heisenberg subalgebras of g, up to conjugacy under

the adjoint action of the Lie group of g (see [12, Proposition of Section 9]). Hence,

the maximal Heisenberg subalgebras of g are parametrized by the conjugacy classes

of W .

Recall that the Weyl group of slr is the symmetric group Sr. Define a partition

of r of length s to be an s-tuple of positive integers (r1, . . . , rs) such that

r1 ≥ · · · ≥ rs, r1 + · · ·+ rs = r.

Every σ ∈ Sr can be written as a product of disjoint cycles, σ = c1 · · · cs, and so σ

defines a partition of r by setting ri equal to the length of the cycle ci (reordering

the ci if necessary). Two elements in Sr are conjugate if and only if they define the

same partition of r. Thus, conjugacy classes in Sr, and hence Heisenberg subalgebras

of slr, are parametrized by partitions of r.

We can extend these ideas to glr (since the Weyl group of glr is also Sr). That is,

to each partition of r we associate a Heisenberg subalgebra of glr as in (1.7) and (1.8).

The most well-known Heisenberg subalgebras of glr are the so-called principal Heisen-

berg subalgebra and the homogeneous Heisenberg subalgebra, which correspond to the

partitions (r) and (1, 1, . . . , 1), respectively. The principal Heisenberg subalgebra is

given by

α(n) =

(E0 + E1 + · · ·+ Er−1)n, for n > 0,

(F0 + F1 + · · ·+ Fr−1)−n for n < 0,

where the Ek and Fk are the Chevalley generators of slr (which, of course, is a sub-

algebra of glr) and exponentiation is meant to be taken in the underlying associative


algebra of glr. Setting

α(0) = I ⊗ 1,

the α(n), n ∈ Z, define a 1-coloured oscillator algebra. The homogeneous Heisenberg

subalgebra, presented as an r-coloured algebra, is given by

α`(n) = E`,` ⊗ tn,

for n ∈ Z− 0 and ` = 1, . . . , r. Setting

α`(0) = E`,` ⊗ 1,

defines an r-coloured oscillator algebra. The Heisenberg subalgebras associated to

the other partitions of r are “blends” of these two extremes. Namely, for an arbitrary

partition (r1, . . . , rs), we divide matrices in glr into s2 blocks, with the (i, j)-th block

of size ri × rj for all i, j = 1, . . . , s. The (`, `)-th diagonal block corresponds to the

subalgebra glr` ⊆ glr, and on the level of affine algebras, glr` ⊆ glr. Let E`k, F

`k , H

`k be

the Chevalley generators for slr` ⊆ glr` . The Heisenberg subalgebra corresponding to

the partition (r1, . . . , rs), presented as an s-coloured algebra, is given by

α`(n) =

(E`0 + E`

1 + · · ·+ E`r`−1)n, for n > 0,

(F `0 + F `

1 + · · ·+ F `r`−1)−n, for n < 0,

for ` = 1, . . . , s. It is worth noting that, restricted to the `-th colour, the α`(n)

determine the principal Heisenberg subalgebra of glr` . Set

α`(0) = I` ⊗ 1,

where I` is the identity matrix in glr` . Then the α`(n) define an s-coloured oscillator

algebra.

To each partition of r, we have associated a Heisenberg (and oscillator) subal-

gebra of glr, and, as mentioned before, each Heisenberg subalgebra gives rise to a


realization of Vbasic. These various realizations were described explicitly in [26] in

terms of vertex operators on bosonic and fermionic Fock space. We summarize the

main theorem of that paper here.

Conceptually, it is easier to begin with the principal realization (corresponding

to the partition (r)) before generalizing to the realizations corresponding to arbitrary

partitions of r. One begins by defining formal fermionic fields

ψ(z) :=∑k∈Z

ψ(k)zk, ψ∗(z) :=∑k∈Z

ψ∗(k)z−k,

where z is a formal variable. These are power series with coefficients in the (1-

coloured) Clifford algebra. Define the normal ordering on the Clifford algebra gener-

ators

: ψ(i)ψ∗(j) : =

ψ(i)ψ∗(j), if j > 0,

−ψ∗(j)ψ(i), if j ≤ 0.

We also define formal bosonic fields

α(z) :=: ψ(z)ψ(z) : .

Expanded as a Laurent series α(z) =∑

k∈Z α(k)z−k, one finds

α(k) =∑i∈Z

: ψ(i)ψ∗(i+ k) : .

Theorem 1.19. Let ω = e2πi/r. Then the homogeneous components of

: ψ(ωpz)ψ∗(ωqz) : − ωp−q

1− ωp−q, 1 ≤ p, q ≤ r, p 6= q,

: ψ(z)ψ∗(z) :,

together with the identity operator, span a Lie algebra of operators on F (1-coloured

fermionic Fock space) that is isomorphic to glr. Moreover, via this identification with

glr,

F(0) ∼= Vbasic,


as glr-modules and the α(k), k ∈ Z−0 (resp. k ∈ Z), together with the identity op-

erator, form a basis of the principal Heisenberg subalgebra (resp. oscillator subalgebra)

of glr.

Proof: This is a special case of Theorem 1.20 below.

Note that the isomorphism F(0)∼=−→ Vbasic in Theorem 1.19 is determined by

|0〉 7→ v0. Also, recall that the idea behind the realizations of Vbasic was to describe

Vbasic in terms of Λ (or bosonic Fock space), whereas Theorem 1.19 describes Vbasic

in terms of fermionic Fock space. However, via the boson-fermion correspondence

(see (1.3)), F(0) ∼= B(0), and so Theorem 1.19 satisfies our goal of describing Vbasic in

terms of bosonic Fock space.

We would now like to generalize Theorem 1.19 to an arbitrary partition of r. We

fix, once and for all, a partition,

r = (r1, . . . , rs),

of r. As with our construction of the Heisenberg subalgebra associated to r, we divide

matrices in glr into s2 blocks of size ri × rj. The operators associated to the (`, `)-th

diagonal blocks correspond to the subalgebra glr` and the `-th colour of the Heisenberg

subalgebra is the principal Heisenberg subalgebra of glr` . Thus, we can simply take s-

copies of the principal realization above, which amounts to taking s-coloured versions

of the Heisenberg and Clifford algebras as well as s-coloured versions of bosonic and

fermionic Fock space. The operators associated to the off-diagonal blocks can be

obtained by “mixing” Clifford algebra generators of different colours.

Let R′ = lcmr1, . . . , rs and define

R =

R′, if R′

(1ri

+ 1rj

)∈ 2Z for all i, j,

2R′, if R′(

1ri

+ 1rj

)/∈ 2Z for some i, j.


Define the normal ordering on the s-coloured Clifford algebra generators

: ψ`(i)ψ∗k(j) : =

ψ`(i)ψ∗k(j), if j > 0,

−ψ∗k(j)ψ`(i), if j ≤ 0.

We introduce the formal s-coloured fermionic and bosonic fields:

ψ`(z) :=∑k∈Z

ψ`(i)z(R/r`)k, ψ∗` (z) :=

∑k∈Z

ψ∗` (k)z−(R/r`)k,

α`(z) =∑k∈Z

α`(k)z−(R/r`)k := : ψ`(z)ψ∗` (z) :,

This brings us to the main theorem of [26] (as stated at the end of the introduction).

Theorem 1.20. Let ω = e2πi/R. Then the homogeneous components of

: ψ`(ωpz)ψ∗k(ω

qz) : −δk`ω(R/r`)(p−q)

1− ω(R/r`)(p−q),

(` 6= k, 1 ≤ p ≤ r`, 1 ≤ q ≤ rk)

or (` = k, 1 ≤ p 6= q ≤ r`),

: ψ`(z)ψ∗` (z) :, 1 ≤ ` ≤ s,

together with the identity operator, span a Lie algebra of operators on F (s-coloured

fermionic Fock space) that is isomorphic to glr. Moreover, via this identification with

glr,

F(0) ∼= Vbasic,

as glr-modules and the α`(k), k ∈ Z − 0 (resp. k ∈ Z), together with the identity

operator, give a basis of the Heisenberg subalgebra (resp. oscillator subalgebra) of glr

associated to the partition r.

As was the case with Theorem 1.19, the isomorphism F(0)∼=−→ Vbasic in Theorem

1.20 is determined by |0〉⊗s 7→ v0.

As stated at the outset, our goal is to give a geometric version of the various

realizations of Vbasic. Our strategy will be as follows. We first construct geometric


analogues the oscillator and Clifford algebra representations on bosonic and fermionic

Fock space. Our method for doing this will be similar to [15, Section 3]. Next, we

would like to construct an action of glr on our geometric fermionic Fock space in the

spirit of Theorem 1.20. To do this we recall a previous observation:

glr = slr ⊕(I ⊗ C[t, t−1]

),

and so glr is generated by the Chevalley generators Ek, Fk, Hk, k = 0, 1, . . . , r − 1

of slr and loops on the identity, I ⊗ tn, n ∈ Z. Thus, to define an action of glr on

our geometric Fock space, it is enough to define the action of Ek, Fk, Hk and I ⊗ tn.

Algebraically, the action of glr on fermionic Fock space is given by the homogeneous

components of vertex operators as in Theorem 1.20. We explicitly describe the action

of Ek, Fk and I⊗tn in the following Lemma 1.21 below (the action of Hk is determined

by Ek and Fk since Hk = [Ek, Fk]). First, we observe that for each k = 0, 1, . . . , r−1,

we can write

k = r1 + · · ·+ r`−1 + k′,

for unique 1 ≤ ` ≤ s and 0 ≤ k′ < r`−1.

Lemma 1.21. The action of glr on F from Theorem 1.20 is given by the map ρ :

glr → gl(F), described below. For each k = 0, 1, . . . , r−1, write k = r1+· · ·+r`−1+k′.

If k′ 6= 0, then

ρ(Ek) =∑i∈Z

ψ`(k′ + ir`)ψ

∗` (k′ + ir` + 1), (1.9)

ρ(Fk) =∑i∈Z

ψ`(k′ + ir` + 1)ψ∗` (k

′ + ir`). (1.10)

If k′ = 0 and ` 6= 1,

ρ(Ek) =∑i∈Z

ψ`−1((i+ 1)r`−1)ψ∗` (ir` + 1), (1.11)

ρ(Fk) =∑i∈Z

ψ`(ir` + 1)ψ∗`−1((i+ 1)r`−1). (1.12)


If k′ = 0 and ` = 1, (i.e. k = 0)

ρ(E0) =∑i∈Z

ψs(irs)ψ∗1(ir1 + 1), (1.13)

ρ(F0) =∑i∈Z

ψ1(ir1 + 1)ψ∗s(irs). (1.14)

Finally,

ρ(I ⊗ tn) =s∑`=1

α`(nr`) =

|n|∑s

`=1 r`P`(nr`), if n 6= 0,∑s`=1 P`(0), if n = 0.

(1.15)

Proof: We will prove Equation (1.9) and leave the remaining equations up to

the reader since they can be derived in a similar manner. The proof simply requires

picking out the appropriate components from the vertex operators constructed in [26].

As before, we decompose the r × r matrices of glr into s2 blocks of size ri × rj. For

1 ≤ i, j ≤ s and 1 ≤ p ≤ ri, 1 ≤ q ≤ rj, define Eijpq to be the matrix with 1 in the

(p, q)-th entry of the (i, j)-th block and zeros elsewhere. Let ωj = e2πi/rj be an rj-th

root of unity. As in [26, Equation (2.3.3)], define

Aijpq :=1√rirj

ri∑a=1

rj∑b=1

ωpai ω−qbj Eij

ab.

The Aijpq’s form a basis of gln. Thus, one can express the Eijpq’s in terms of the Aijpq’s

by [26, Equation (2.3.7)],

Eijpq =

1√rirj

ri∑a=1

rj∑b=1

ω−pai ωqbj Aijab. (1.16)

Now, as in [26, Equation (3.3.4)], define

hr =s∑`=1

r∑p=1

r` − 2p+ 1

2r`E``pp.

The element hr induces a ZR-gradation of glr,

glr =⊕n∈ZR

gl(n)r , gl(n)

r := x ∈ glr | [hr, x] =n+mR

Rx for some m ∈ Z,


where we use the notation n = n mod R. Then the elements

x(n) := e−iθ adhrei(n/R)θxn, x = (xn) ∈ glr, n ∈ Z, (1.17)

form a spanning set of glr (see [26, Equation (4.2.4)]). For each x ∈ glr and n ∈ Z,

define

x(n) := x(n)− δn0 tr(hrx)c, (1.18)

as in [26, Equation (4.2.9)]. The map ρ : glr → gl(F) is given explicitly in terms of

the elements Aijpq(n). Namely, we define the formal power series

Aijpq(z) :=∑n∈Z

ρ(Aijpq(n))z−n,

and let ρ be the map determined by setting

Aijpq(z) =z(R/2)((1/rj)−(1/ri))

√rirj

: ψi(ωpz)ψ∗j (ω

qz) : − 1

riδij

ω(R/ri)(p−q)

1− ω(R/ri)(p−q), if p 6= q,

(1.19)

Aijpp(z) :=z(R/2)((1/rj)−(1/ri))

√rirj

: ψi(ωpz)ψ∗j (ω

pz) : . (1.20)

Note that our equations vary slightly from [26, Equation 5.5.4] since we do not “ab-

sorb” the zR/2ri terms into our definition of the fermionic fields.

Let k = r1 + · · ·+ r`−1 + k′ ∈ 0, 1, . . . , r − 1 such that k′ 6= 0. Then

Ek = Ek,k+1 = E``k′,k′+1.

By [26, Equation (3.4.5)],

[hr, E``k′,k′+1] =

(k′ + 1

r`− k′

r`+

1

2r`− 1

2r`

)E``k′,k′+1 =

1

r`E``k′,k′+1,

and so E``k′,k′+1 ∈ gl(R/r`)r and, by Equations (1.17), (1.18) and (1.16),

Ek = E``k′,k′+1(R/r`) = E``

k′,k′+1(R/r`) =1

r`

r∑p,q=1

ω−(p−q)k′+q` A``pq(R/r`).


Now, ρ(A``pq(R/r`)) is equal to the coefficient of z−R/r` in the vertex operators in

Equations (1.19) and (1.20). A quick calculation then reveals that

ρ(A``pq(R/r`)) =1

r`

∑i∈Z

ω(p−q)i−q` ψ`(i)ψ

∗` (i+ 1).

Thus,

ρ(Ek) =1

r2`

r∑p,q=1

∑i∈Z

ω(i−k′)(p−q)` ψ`(i)ψ

∗` (i+1) =

1

r`

∑i∈Z

r`−1∑(p−q)=0

ω(i−k′)(p−q)` ψ`(i)ψ

∗` (i+1).

Since,

r`−1∑(p−q)=0

ω(i−k′)(p−q)` =

r`, if r` | (i− k′),

0, otherwise,

one finds,

ρ(Ek) =∑

r`|(i−k′)

ψ`(i)ψ∗` (i+ 1) =

∑i∈Z

ψ`(k′ + r`i)ψ

∗` (k′ + r`i+ 1).

Chapter 2

Geometric Invariant Theory

Our main geometric objects of interest will be the so-called Nakajima quiver varieties,

which are examples of geometric (invariant) quotients. The purpose of this chapter

will be to review some of the basic results from geometric invariant theory that we

will need before studying quiver varieties in the following chapter. The material in

this chapter is based largely on [19]. However, while [19] deals in full generality with

schemes over an arbitrary field, we will deal exclusively with algebraic varieties over

C.

Throughout this chapter and all subsequent chapters, whenever X is a set and

G is a group acting on X, we will denote the set of G-fixed points of X by XG.

Let X be an algebraic variety and let G an algebraic group acting on X, i.e. the

group action

σ : G×X → X,

(g, x) 7→ g · x,

is a morphism of varieties. Then for any G-invariant open set U ⊆ X, we have an

induced action of G on OX(U) given by

(g · f)(x) = f(g−1 · x),

28

2. Geometric Invariant Theory 29

for all g ∈ G, f ∈ OX(U) and x ∈ U . Note that if f ∈ OX(U)G, then f is constant

on the orbit G · x for each x ∈ U . Thus, f induces a well-defined function on the

orbit space U/G given by

f : U/G→ C,

G · x 7→ f(x).(2.1)

This leads us to the definition of a geometric quotient. Note that our definition is a

slightly simplified version of [19, Definition 0.6].

Definition 2.1 (Geometric quotient). Let X be an algebraic variety and let G an

algebraic group acting on X with group action σ : G × X → X. A pair (Y, φ),

where Y is an algebraic variety and φ : X → Y is a morphism of varieties, is called a

geometric quotient of X by G if the following conditions hold.

1. The diagram

G×X X

X Y

σ

p

φ

φ,

commutes, where the map p is the projection onto the second factor.

2. The map φ is surjective and the fibres of φ are precisely the G-orbits of X (i.e.

for all y ∈ Y , there exists an x ∈ X such that φ−1(y) = G · x).

3. The variety Y has the quotient topology induced by φ.

4. The structure sheaf of Y is given by

OY (U) =f : U → C | f ∈ OX(φ−1(U))G

,

where f is defined as in Equation (2.1).


Remark 2.2. Geometric quotients are not guaranteed to exist. However, if a geo-

metric quotient (Y, φ) of X by G does exist, then by (2) and (3) of Definition 2.1, it

is clear that, topologically, Y ∼= X/G. Hence, whenever a geometric quotient (Y, φ)

exists, we will identify it with (X/G, π), where π : X X/G is the canonical pro-

jection. Moreover, by a slight abuse of notation, we will simply refer to X/G as the

geometric quotient of X by G, leaving the map π implied.

We will, of course, also be interested in morphisms between geometric quotients.

The next lemma will prove useful in constructing such morphisms.

Lemma 2.3. Let X and Y be algebraic varieties and let GX and GY be algebraic

groups acting on X and Y , respectively, such that X/GX and Y/GY are geometric

quotients. If ϕ : X → Y is a morphism of varieties and ψ : GX → GY is a group

homomorphism such that

GY × Y Y

GX ×X X

ψ × ϕ ϕ , (2.2)

commutes (where the horizontal arrows represent the group action), then the induced

map

ϕ : X/GX → Y/GY ,

GX · x 7→ GY · ϕ(x),

is a well-defined morphism of varieties. Moreover, if ϕ and ψ are isomorphisms of

varieties and groups, respectively, then ϕ is an isomorphism of varieties.

Proof: The fact that ϕ is well-defined follows directly from the commutativity

of Diagram (2.2). Clearly, ϕ is continuous. Thus, to show that ϕ is a morphism of

varieties, it remains only to show that the mapping f 7→ f ϕ is a pullback

OY/GY (U)→ OX/GX (ϕ−1(U)),


for all open U ⊆ Y . Let πX : X X/GX and πY : Y Y/GY denote the canonical

projections. By definition of geometric quotients, this is equivalent to showing that

the mapping f 7→ f ϕ is a map

OY (π−1Y (U))GY → OX(π−1

X (ϕ−1(U))GX = OX(ϕ−1(π−1Y (U))GX ,

for all open U ⊆ Y . For any f ∈ OY (π−1Y (U))GY , x ∈ π−1

X (ϕ−1(U)) and g ∈ GX ,

(g · (f ϕ))(x) = f(ϕ(g−1 · x)) = f(ψ(g−1) · ϕ(x)) = f(ϕ(x)) = (f ϕ)(x),

where the second equality follows from Diagram (2.2) and the third equality follows

from the fact that f ∈ OY (π−1Y (U))GY . Therefore, we conclude that ϕ is a morphism

of varieties.

In the case that ϕ and ψ are isomorphisms, we have the following commutative

diagram:

GY × Y Y

GX ×X X

ψ−1 × ϕ−1 ϕ−1.

By repeating all the same arguments, we have a morphism of varieties

ϕ−1 : Y/GY → X/GX ,

GY · y 7→ GX · ϕ−1(y),

which is the inverse of ϕ. Thus, ϕ is an isomorphism.

We now turn our attention to tangent spaces, as they offer a very useful means

for studying the local behaviour of a variety. We begin by recalling the definition.

Let X be an algebraic variety. Recall that the stalk of x ∈ X is

OX,x := lim−→OX(U) (for open sets U 3 x)


= [f, U ] | open U 3 x and f ∈ OX(U),

where [f, U ] is the equivalence class of (f, U) under the equivalence relation (f, U) ∼

(f ′, U ′) if and only if there exists an open V ⊆ U∩U ′ such that x ∈ V and f |V = f ′|V .

By a slight abuse of notation, we will denote [f, U ] simply by f , leaving the open set

U implied. A derivation of OX,x is a linear map δ : OX,x → C such that

δ(fg) = δ(f)g(x) + f(x)δ(g),

for all f, g ∈ OX,x.

Definition 2.4. (Tangent space) Let X be an algebraic variety. For x ∈ X, the

tangent space of X at x is the set of derivations of OX,x, i.e.

Tx(X) := δ ∈ HomC(OX,x,C) | δ(fg) = δ(f)g(x) + f(x)δ(g), for all f, g ∈ OX,x.

If X is an affine variety, we may define the tangent space in more global terms

by replacing the stalk OX,x with the coordinate ring C[X] of X, i.e.

Tx(X) = δ ∈ HomC(C[X],C) | δ(fg) = δ(f)g(x) + f(x)δ(g), for all f, g ∈ C[X].

If X and Y are algebraic varieties and ϕ : X → Y is a morphism of varieties,

then we have an induced map on the stalks

ϕ∗ : OY,ϕ(x) → OX,x,

f 7→ f ϕ,

for each x ∈ X (in the affine case, we may replace the stalks OX,x and OY,ϕ(x) with

the coordinate rings C[X] and C[Y ]). This induces a map on the tangent spaces

dϕ : Tx(X)→ Tϕ(x)(Y ),

δ 7→ δ ϕ∗,


called the differential of ϕ (at x). If X is smooth, then we say that ϕ is etale if dϕ

is an isomorphism. Note that this is not the original definition of an etale morphism

(see [7, Definition 17.3.1]), but rather an equivalent characterization in the setting of

smooth varieties (see [7, Corollary 17.11.2]).

Let G be an algebraic group acting on X and let σ : G × X → X denote the

group action. Define σg := σ(g,−) : X → X and σx := σ(−, x) : G → X for all

g ∈ G and x ∈ X. We then have induced maps on the tangent spaces

dσg : Tx(X)→ Tg·x(X) and dσx : T1(G)→ Tx(X).

For sufficiently nice varieties, the following lemma allows us to compute the tangent

space of X/G.

Lemma 2.5. Let X be a smooth algebraic variety and let G be a reductive algebraic

group acting freely on X such that the geometric quotient X/G exists. Let π : X

X/G denote the canonical projection. Then, for all x ∈ X,

0→ T1(G)dσx−−→ Tx(X)

dπ−→ TG·x(X/G)→ 0,

is a short exact sequence.

Proof: Luna’s etale slice theorem (see [16, Section III, Theoreme du slice etale])

implies that, for all x ∈ X, there exists a subvariety V ⊆ X, which we may assume to

be smooth (see Remark 1 following the slice theorem in [16]), containing x such that

σ|G×V is etale (note that since G acts freely on X, the stabilizer Gx is trivial and the

orbit G · x is closed). Therefore,

dσ : T(1,x)(G× V )→ Tx(X),

is an isomorphism. Luna’s etale slice theorem also implies that the morphism σ,

induced from the following commutative diagram


G× V X

(G× V )/G ∼= V X/G,

σ

σ

where the vertical maps are the canonical projections, is etale. Thus,

dσ : TG·(1,x)((G× V )/G ∼= Tx(V )→ TG·x(X/G),

is an isomorphism. Notice that we have a commutative diagram:

Gi1

−−−→ G× Vπσ−−−− X/G

id

y σ

y id

yG

σx

−−−→ Xπ−− X/G,

where i1(g) = (g, x) for all g ∈ G. This induces the following commutative diagram

on the tangent spaces:

0 → T1(G)di1

−−−−→T(1,x)(G× V )

∼= T1(G)⊕ Tx(V )

dπdσ−−−−− TG·x(X/G) → 0

id

y dσ

y id

y0 → T1(G)

dσx

−−−−→ Tx(X)dπ−−− TG·x(X/G) → 0.

(2.3)

Clearly, the first row of Diagram (2.3) is a short exact sequence. Since the vertical

arrows of Diagram (2.3) are all isomorphisms, it follows that the second row is also a

short exact sequence.

Corollary 2.6. Let X and G be as in Lemma 2.5. Then, for all x ∈ X,

TG·x(X/G) ∼= Tx(X)/ im dσx.

Proof: By Lemma 2.5 and the first isomorphism theorem, TG·x(X/G) = im dπ ∼=

Tx(X)/ ker dπ = Tx(X)/ im dσx.


Let us keep the assumptions of Lemma 2.5. Suppose ϕ : X → X is a G-

equivariant morphism of varieties. Then we have a well-defined morphism

ϕ : X/G→ X/G,

G · x 7→ G · ϕ(x).

Let x ∈ X such that G · x ∈ X/G is a fixed point of ϕ. Then, there exists a unique

g ∈ G such that ϕ(x) = g−1 · x and the map ϕ induces a map on TG·x(X/G):

dϕ : TG·x(X/G)→ TG·x(X/G).

By Corollary 2.6, we may identify TG·x(X/G) with Tx(X)/ im dσx. Via this iden-

tification, dϕ induces a map on Tx(X)/ im dσx, which we compute in the following

lemma.

Lemma 2.7. Let X, G, ϕ, x and g be as above. Let

ψ : TG·x(X/G)→ Tx(X)/ im dσx,

be the isomorphism described in Corollary 2.6. Then we have well-defined maps

dϕ : Tx(X)/ im dσx → Tϕ(x)(X)/ im dσϕ(x),

δ + im dσx 7→ dϕ(δ) + im dσϕ(x),

and

dσg : Tϕ(x)(X)/ im dσϕ(x) → Tx(X)/ im dσx,

δ + im dσϕ(x) 7→ dσg(δ) + im dσx.


Moreover, the following diagram commutes.

TG·x(X/G) TG·x(X/G)

Tx(X)/ im dσx

Tϕ(x)(X)/ im dσϕ(x)

Tx(X)/ im dσx

dϕ

ψ ψ

dϕ dσg

(2.4)

In particular, the induced action of dϕ on Tx(X)/ im dσx is given by the dashed line

in Diagram (2.4)

Proof: Let τg : G→ G be the map τg(h) = ghg−1. Note that τg is a morphism of

varieties. We have the following commutative digram:

Gσx

−−−→ Xπ−− X/G

id

y ϕ

y ϕ

yG

σϕ(x)

−−−−→ Xπ−− X/G

τg

y σg

y id

yG

σx

−−−→ Xπ−− X/G,

which induces the following commutative diagram on the tangent spaces:

0 → T1(G)dσx

−−−−→ Tx(X)dπ−−− TG·x(X/G) → 0

id

y dϕ

y dϕ

y0 → T1(G)

dσϕ(x)

−−−−→ Tϕ(x)(X)dπ−−− TG·x(X/G) → 0

dτg

y dσg

y id

y0 → T1(G)

dσx

−−−−→ Tx(X)dπ−−− TG·x(X/G) → 0,

where every row is a short exact sequence by Lemma 2.5. Therefore, dϕ(im dσx) ⊆

im dσϕ(x) and dσg(im dσϕ(x)) ⊆ im dσx, thus proving that dϕ and dσg are well-defined.


One then easily verifies the commutativity of Diagram (2.4).

Remark 2.8. Let X and G be as in Lemma 2.7 and let H be an algebraic group

acting on X such that the G and H actions commute. Then each h ∈ H induces a

G-equivariant morphism of varieties ϕh : X → X, given by x 7→ h · x. Let x ∈ X

such that G · x ∈ (X/G)H . Then H acts on TG·x(X/G) by

h · δ = dϕh(δ),

for all h ∈ H and δ ∈ TG·x(X/G). By Corollary 2.6, we may identify TG·x(X/G) with

Tx(X)/ im dσx. For each h ∈ H, let g(h) ∈ G be the element such that h·x = g(h)−1·x.

By Lemma 2.7, the induced action of H on Tx(X)/ im dσx is given by

h · (δ + im dσx) = (dσg(h) dϕh)(δ + im dσx) =(dσg(h) dϕh

)(δ) + im dσx,

for all h ∈ H and δ + im dσx ∈ Tx(X)/ im dσx.

Let X be an algebraic variety (not necessarily smooth) and G an algebraic group

(not necessarily reductive) acting on X. Let g ∈ G and consider the morphism

φg : X → X ×X,

x 7→ (x, g · x).

By definition of a variety, the diagonal

∆(X) = (x, x) ∈ X ×X,

is a closed subset X ×X. Therefore,

φ−1g (∆(X)) = Xg,

is closed in X. Hence,

XG =⋂g∈G

Xg,


is a closed subvariety of X. For each g ∈ G, let

ϕg : X → X,

x 7→ g · x.

Recall that, for each x ∈ XG, the group G acts on Tx(X) via

g · δ = dϕg(δ),

for all g ∈ G and δ ∈ Tx(X). The following lemma shows that we may naturally

identify Tx(XG) with Tx(X)G.

Lemma 2.9. Let X be an algebraic variety and let G be an algebraic group acting on

X. Let

i : XG → X,

be the inclusion map. Then, for all x ∈ X, the differential di maps Tx(XG) isomor-

phically onto Tx(X)G.

Proof: Without loss of generality, we may assume that X is affine. Suppose

X ⊆ An and let C[X] and C[XG] be the coordinate rings of X and XG, respectively.

Both C[X] and C[XG] are quotients of the polynomial ring C[t1, . . . , tn]. For each

g ∈ G, we may consider the morphism ϕg to be an n-tuple (ϕ1g, . . . , ϕ

ng ) of polynomials

in C[X]. Then

C[XG] ∼= C[X]/J,

where J is the radical of the ideal generated by the set ϕig− ti | g ∈ G, i = 1, . . . , n.

In particular, ϕig+J = ti+J in C[XG] and so, ϕ∗g(f)+J = f+J for all f+J ∈ C[XG].

The pullback of i is the map

i∗ : C[X] C[XG],

f 7→ f + J.


Now, we consider the differential di : Tx(XG) → Tx(X). Since i∗ is surjective, it is

clear that di is injective. Thus, it remains only to show that di(Tx(XG)) = Tx(X)G.

Let δ ∈ Tx(XG). Then for any g ∈ G and f ∈ C[X],

(g · (di(δ)))(f) = δ(ϕ∗g(f) + J) = δ(f + J) = (di(δ))(f).

Thus, di(δ) ∈ Tx(X)G and so, di(Tx(XG)) ⊆ Tx(X)G. Conversely, let ε ∈ Tx(X)G.

Notice that, for all g ∈ G and i ∈ 1, . . . , n,

ε(ϕig − ti) = ε(ϕig)− ε(ti) = (g · ε)(ti)− ε(ti) = ε(ti)− ε(ti) = 0.

Thus, ε vanishes on the generators of J . Moreover, the generators of J vanish on x,

and so ε(J) = 0. Hence, we have a well-defined derivation

δ : C[XG]→ C,

f + J 7→ ε(f),

and di(δ) = ε. Therefore, Tx(X)G ⊆ di(Tx(XG)), completing the proof.

In the following chapters, we will commonly identify the tangent space of varieties

with the middle cohomology of certain complexes. In light of the previous lemma,

it will be useful for us to compute the fixed points of the middle cohomology of

complexes.

Lemma 2.10. Let G be a finite-dimensional torus or a finite group and let

Xα

−−−→ Yβ−− Z

be a complex of of G-modules such that α is injective and β is surjective. Let αG =

α|XG and βG = β|Y G. Then ker βG/ imαG ∼= (ker β/ imα)G as G-modules.


Proof: We have the following commutative diagram

XG αG−−−−→ Y G

βG−−− ZG

→

i

→ →

Xα

−−−→ Yβ−− Z,

where the vertical arrows represent the inclusion maps. Note that the restrictions

αG and βG remain injective and surjective, respectively. By commutativity of the

diagram, i induces a G-module morphism

i∗ : ker βG/ imαG → ker β/ imα,

y + imαG 7→ i(y) + imα = y + imα.

Suppose y+imαG ∈ ker i∗. This implies that y ∈ imα, and so we have a (unique)

x ∈ X such that α(x) = y. Because y ∈ Y G, for every g ∈ G we have

α(g · x) = g · α(x) = g · y = y = α(x).

Since α is injective, we have that g · x = x. Therefore, x ∈ XG and so y ∈ imαG.

Thus ker i∗ = 0 and i∗ is injective. Hence, by the First Isomorphism Theorem,

ker βG/ imαG ∼= im i∗.

We claim that im i∗ = (ker β/ imα)G, which would complete the proof of the

lemma. Clearly, im i∗ ⊆ (ker β/ imα)G, thus it remains only to check the reverse

inclusion. If G = (C∗)d is a d-dimensional torus, then we have a weight space decom-

position of X:

X =⊕k∈Zd

Xk, Xk := x ∈ X | (g1, . . . , gd)·x = gk11 · · · g

kdd x, for all (g1, . . . , gd) ∈ G.

We also have a weight space decomposition Y =⊕

k∈Zd Yk, where the Yk are defined in

the same way as the Xk. Suppose y = (yk) ∈ ker β such that y+imα ∈ (ker β/ imα)G.

Thus, for every g ∈ G, we have (g · y) − y ∈ imα. Moreover, since α is a morphism

of G-modules, α|Xk: Xk → Yk, and so

((g · y)− y)k = (gk11 · · · g

kdd − 1)yk ∈ imα,


for all k ∈ Zd and g = (g1, . . . , gd) ∈ G. For each k 6= 0, we may choose a g =

(g1, . . . , gd) ∈ G such that gk1 · · · gkd 6= 1, and thus yk ∈ imα. Therefore,

y + imα = y0 + imα ∈ im i∗,

since y0 ∈ Y G.

IfG is a finite group, we again suppose y ∈ ker β such that y+imα ∈ (ker β/ imα)G,

i.e. (g · y)− y ∈ imα for all g ∈ G. Thus,

1

|G|∑g∈G

(g · y − y) =

(1

|G|∑g∈G

g · y

)− y ∈ imα,

and so

y + imα =1

|G|∑g∈G

g · y + imα ∈ im i∗,

since 1|G|∑

g∈G g · y ∈ Y G.

We end this chapter with the following lemma, which gives a sufficient set of

conditions to determine when a morphism of varieties is an isomorphism.

Lemma 2.11. Let X and Y be smooth algebraic varieties and let ϕ : X → Y be a

bijective etale morphism. Then ϕ is an isomorphism (of varieties).

Proof: The Inverse Function Theorem found in [18, Theorem 5.31] implies that ϕ

is birational. Then, since ϕ is bijective and Y is smooth (in particular, Y is normal),

by a version of Zariski’s Main Theorem (see [20, Chapter III, Section 9, Theorem I]),

it follows that ϕ is an isomorphism.

Chapter 3

Quiver Varieties

The key result of this chapter is Theorem 3.20, in which we show that the Zm-fixed

point set of certain Nakajima quiver varieties is isomorphic to a disjoint union of

Nakajima quiver varieties of type Am−1. In Chapter 5, we will see that the equivariant

cohomology of these varieties will provide us with a suitable geometric analogue of

Vbasic.

Let Q = (Q0, Q1) be a quiver. For all ρ ∈ Q1, write t(ρ) and h(ρ) for the tail

and the head of ρ, respectively. Let Q = (Q0, Q1) be the double quiver of Q. That is,

Q0 = Q0, and Q1 = Q1 ∪Q1,

where Q1 is the set of arrows in Q1 with orientation reversed. We then have a natural

involution ¯ : Q1 → Q1, which sends every arrow in Q1 to its corresponding reverse

arrow in Q1 and vice versa. We define the function ε : Q1 → ±1 by

ε(ρ) =

1, if ρ ∈ Q1,

−1, if ρ ∈ Q1.

Let V =⊕

k∈Q0Vk and W =

⊕k∈Q0

Wk be Q0-graded complex vector spaces and let

42

3. Quiver Varieties 43

n = dimV and s = dimW . Define

EV =⊕ρ∈Q1

Hom(Vt(ρ), Vh(ρ)), LV,W =⊕k∈Q0

Hom(Vk,Wk).

We define a “multiplication” EV × EV → LV,V given by

(AB)k =∑

ρ∈Q1,t(ρ)=k

AρBρ,

for all A = (Aρ), B = (Bρ) ∈ EV . We then define

M = M(V,W ) = EV ⊕ LW,V ⊕ LV,W .

The function ε : Q1 → ±1 induces a function ε : EV → EV given by

ε(C)ρ = ε(ρ)Cρ.

We have a symplectic form ω on M given by

ω((C1, i1, j1), (C2, i2, j2)) = tr(ε(C1)C2) + tr(i1j2 − i2j1).

Let GV =∏

k∈Q0GL(Vk). Then GV acts on M via

g · (C, i, j) = (gCg−1, gi, jg−1),

for all g ∈ GV and (C, i, j) ∈M. Then GV is an algebraic group whose action on M

preserves the symplectic form ω. The moment map vanishing at the origin is given

by

µ : M→ LV,V ,

(C, i, j) 7→ ε(C)C + ij,

where the Lie algebra LV,V of GV is identified with its dual via the trace.

Remark 3.1. Note that the set M and the map µ do not depend on the orientation

of the quiver Q. Thus, our construction above, as well as the definition of quiver

varieties below, depend only on the underlying (undirected) graph of Q.


Definition 3.2 (Invariant, Stable). Let S =⊕

k∈Q0Sk, where each Sk is a subspace

of Vk. For C ∈ EV , we say that S is C-invariant if Cρ(St(ρ)) ⊆ Sh(ρ) for all ρ ∈ Q1.

An element (C, i, j) ∈M is called stable if the following condition holds: if S is

a Q0-graded subspace of V such that S is C-invariant and im(i) ⊆ S, then S = V .

We denote by Mst the set of stable points of M.

Remark 3.3. Recall that a path (of length `) in a quiver is a sequence of arrows

ρ` · · · ρ2ρ1 such that h(ρk) = t(ρk+1). For any path p = ρ` · · · ρ2ρ1 and any C ∈ EV ,

let C(p) := Cρ` · · ·Cρ2 Cρ1 , which is a linear map Vt(ρ1) → Vh(ρ`). For (C, i, j) ∈M,

define

S(C, i, j) := spanC(p)i(xk) | p is a path starting at k, xk ∈ Wk, k ∈ Q0

.

Notice that (C, i, j) is stable if and only if S(C, i, j) = V . Indeed, suppose (C, i, j) is

stable. We have that S(C, i, j) is a Q0-graded subspace with k-th component Sk :=

S(C, i, j)∩Vk for all k ∈ Q0. Moreover, S(C, i, j) is C-invariant and im(i) ⊆ S(C, i, j),

thus S(C, i, j) = V .

Conversely, suppose S(C, i, j) = V . Let T be a C-invariant Q0-graded subspace

of V with im(i) ⊆ T . Then we have S(C, i, j) ⊆ T , and so T = V . Thus, (C, i, j) is

stable.

Lemma 3.4. The set Mst0 := µ−1(0) ∩Mst is a quasi-affine algebraic variety.

Proof: We will show that Mst (resp. µ−1(0)) is a Zariski open (resp. closed) subset

of M. By choosing bases for Vk and Wk for each k ∈ Q0, we may identify M with a

direct sum of matrix vector spaces, which can then be identified with affine space. Via

this identification, it is clear that µ−1(0) is the vanishing set of a set of polynomials

whose variables are the entries of the matrices defined by C, i and j, and thus µ−1(0)

is a Zariski closed set.

By Remark 3.3,

Mst = (C, i, j) ∈M | S(C, i, j) = V .


Suppose x ∈ Wk, for some k ∈ Q0, such that i(x) 6= 0, and p = ρm · · · ρ2ρ1 is a path in

Q starting at k with m ≥ n = dimV . Let p` = ρ` · · · ρ2ρ1, for all 1 ≤ ` ≤ m. Choose

` minimal such that i(x), C(p1)i(x), . . . , C(p`)i(x) is linearly dependent. Then

C(p`)i(x) = a0i(x) + a1C(p1)i(x) + · · ·+ a`−1C

(p`−1)i(x),

for some aq ∈ C, q = 0, 1, . . . , ` − 1, where we may choose aq = 0 if h(ρq) 6= h(ρ`).

Therefore,

C(p)i(x) = (Cρm · · · Cρ`+1)(a0i(x) + a1C

(p1)i(x) + · · ·+ a`−1C(p`−1)i(x)),

and so C(p)i(x) ∈ spanC(p′)i(x) | p′ is a path of length smaller than m. Thus, by

induction,

S(C, i, j) = spanC(p)i(xk) | p is a path starting at k

of length less than n, xk ∈ Wk, k ∈ Q0.

For each k ∈ Q0 let i(k)` , 1 ≤ ` ≤ dimWk, denote the images under i of the basis

elements of Wk. Let X(C, i, j) be the matrix whose columns are C(pk)i(k)`k

, where we

take all the paths pk starting at k of length less that n, 1 ≤ `k ≤ dimWk, k ∈ Q0. By

the above argument, S(C, i, j) is the column space of X(C, i, j), and so, by basic linear

algebra, S(C, i, j) = V if and only if rk(X(C, i, j)) = n. The rank of a matrix may

be defined as the size of the largest (square) submatrix with nonzero determinant.

Hence, Mst may be described as

Mst = (C, i, j) ∈M | there is an n× n submatrix of X(C, i, j) with det 6= 0.

The complement of Mst, which we denote Munst (the set of unstable points), is thus

Munst = (C, i, j) ∈M | every n× n submtarix of X(C, i, j) has det = 0.

Under our identification with affine space, the determinant of a submatrix yields a

polynomial whose variables are the entries of the submatrix, hence Munst is a vanishing


set of polynomials, and thus a Zariski closed set. Therefore, Mst is a Zariski open

set.

Therefore, since Mst0 is the intersection of an open subset and a closed subset of

M, it is a quasi-affine variety.

Lemma 3.5. The group GV acts freely on Mst.

Proof: Let (C, i, j) ∈Mst. Suppose g = (gk) ∈ GV such that g ·(C, i, j) = (C, i, j)

(i.e. gCg−1 = C, gi = i, and jg−1 = j) and let v ∈ V . By Remark 3.3, S(C, i, j) = V ,

thus we may write v as a finite sum:

v =∑q

cqC(pq)i(xkq),

where cq ∈ C, pq is a path in Q starting at kq, xkq ∈ Wkq , and kq ∈ Q0, for each q.

Applying g to v we get

g(v) = g

(∑q

cqC(pq)i(xkq)

)=∑q

cqgC(pq)i(xkq) =

∑q

cqC(pq)gi(xkq)

=∑q

cqC(pq)i(xkq) = v.

Therefore, g = 1 and thus GV acts freely on Mst.

Definition 3.6 (Nakajima Quiver Variety). The Nakajima quiver variety (associated

to Q) is the geometric quotient

M = M(V,W ) := Mst0 (V,W )/GV .

We will write [C, i, j]GV (or simply [C, i, j] when there is no risk of confusion) to

denote the GV -orbit of a point (C, i, j) ∈Mst0 .


As stated in Remark 2.2, geometric quotients are not guaranteed to exist, thus

Defintion 3.6 requires some justification. The fact that there exists a geometric quo-

tient of Mst0 by GV is essentially proved by Nakajima in [21, Corollary 3.12], though

our definition of quiver varieties in Definition 3.6 is seemingly different from Naka-

jima’s definition in [21, Section 3.ii]. For convenience, we make precise the relation

between the two definitions.

Definition 3.7 (Costable). An element (C, i, j) ∈ M is costable if whenever a Q0-

graded C-invariant subspace S ⊆ V satisfies j(S) = 0, then S = 0. Denote by Mcost

the set of costable points of M and by Mcost0 the intersection Mcost ∩ µ−1(0).

Remark 3.8. Using arguments completely analogous to Lemmas 3.4 and Lemma

3.5, one can easily show that Mcost0 is a quasi-affine variety and GV acts freely on

Mcost0 .

Definition 3.9 (Dual Nakajima quiver variety). The dual Nakajima quiver variety

(associated to Q) is the quotient

M∗ = M∗(V,W ) := Mcost0 (V,W )/GV .

Remark 3.10. Note that in [21, Definition 3.9], Definition 3.7 is used as the definition

of stability. By [21, Corollary 3.12], the Nakajima quiver variety defined in [21, Section

3.ii] corresponds to our definition of the dual Nakajima quiver variety in Definition 3.9.

However the conditions of stability (in Definition 3.2) and costability (in Definition

3.7) are merely duals of each other, which yields an isomorphism of varieties between

Mst0 and Mcost

0 . This in turn will allow us to naturally identify M with M∗.

Lemma 3.11. There exists an isomorphism of varieties Mst0 (V,W )→Mcost

0 (V ∗,W ∗)

and a group isomorphism GV → GV ∗ such that the following diagram commutes:


GV ∗ ×Mcost0 (V ∗,W ∗) Mcost

0 (V ∗,W ∗)

GV ×Mst0 (V,W ) Mst

0 (V,W )

,

where the horizontal arrows represent the group action.

Proof: Let (C, i, j) ∈M. For each Cρ ∈ Hom(Vt(ρ), Vh(ρ)), we have the transpose

map (Cρ)∗ ∈ Hom(V ∗h(ρ), V

∗t(ρ)), so we have a mapping EV → EV ∗ given by C 7→ C∗,

where

(C∗)ρ = (Cρ)∗,

for every ρ ∈ Q1. Moreover, since i ∈ LW,V and j ∈ LV,W , we have i∗ ∈ LV ∗,W ∗ and

j∗ ∈ LW ∗,V ∗ . Hence, we have a map

M(V,W )→M(V ∗,W ∗),

(C, i, j) 7→ (C∗, j∗, i∗).(3.1)

We claim that above mapping induces a map M(V,W ) → M∗(V ∗,W ∗). First, let

(C, i, j) ∈Mst0 . Note that ε(C∗)C∗ = C∗ε(C)∗. Indeed, for every ρ ∈ Q1,

ε(C∗)ρ(C∗)ρ = ε(ρ)(C∗)ρ(C

∗)ρ = ε(ρ)(Cρ)∗(Cρ)

∗ = (Cρ)∗(ε(C)ρ)

∗ = (C∗)ρ(ε(C)∗)ρ.

Thus, for each k ∈ Q0, the Hom(V ∗k , V∗k ) component of ε(C∗)C∗ is

(ε(C∗)C∗)k =∑

ρ∈Q1, t(ρ)=k

ε(C∗)ρ(C∗)ρ =

∑ρ∈Q1, t(ρ)=k

(C∗)ρ(ε(C)∗)ρ = (C∗ε(C)∗)k.

Hence, applying the moment map to (C∗, j∗, i∗) gives

µ(C∗, j∗, i∗) = ε(C∗)C∗ + j∗i∗ = C∗ε(C)∗ + j∗i∗ = (ε(C)C + ij)∗ = µ(C, i, j)∗ = 0,

where the final equality holds because (C, i, j) ∈ µ−1(0). Next we show that (C∗, j∗, i∗)

is costable. Indeed, suppose S =⊕

k∈Q0Sk ⊆ V ∗ such that S is C∗-invariant and


i∗(S) = 0. For each k ∈ Q0, let Uk = v ∈ Vk | ϕ(v) = 0 for all ϕ ∈ Sk and set

U =⊕

k∈Q0Uk. For any ρ ∈ Q1, u ∈ Ut(ρ) and ϕ ∈ Sh(ρ) we have

ϕ(Cρ(u)) = (Cρ)∗(ϕ)(u) = (C∗)ρ(ϕ)︸︷︷︸

∈St(ρ)

(u) = 0.

Therefore, Cρ(u) ∈ Uh(ρ), and so U is C-invariant. Moreover, for any i(x) ∈ im(i)

and ϕ ∈ S,

ϕ(i(x)) = i∗(ϕ)︸︷︷︸=0

(x) = 0.

Thus, i(x) ∈ U , and so im(i) ⊆ U . Because (C, i, j) is stable, we have that U = V .

By construction of U , ϕ(V ) = ϕ(U) = 0 for every ϕ ∈ S, which means S = 0. Hence,

(C∗, j∗, i∗) is costable.

Therefore, the mapping (C, i, j) 7→ (C∗, j∗, i∗) is a mapping

Mst0 (V,W )→Mcost

0 (V ∗,W ∗).

Note that this mapping is a morphism of varieties (since by identifying M with affine

space, this map corresponds to permuting coordinates and is thus a regular map).

Furthermore, by a similar argument, one can show that the mapping

Mcost0 (V ∗,W ∗)→Mst

0 (V ∗∗,W ∗∗),

(C, i, j) 7→ (C∗, j∗, i∗),(3.2)

is also a morphism of varieties. Via the canonical isomorphisms V ∗∗ ∼= V and W ∗∗ ∼=

W , one has a natural isomorphism of varieties

Mst0 (V ∗∗,W ∗∗)

∼=−→Mst0 (V,W ). (3.3)

The composition of the maps (3.2) and (3.3) is the inverse of map (3.1). Thus, the

map (3.1) is an isomorphism of varieties Mst0 (V,W )

∼=−→Mcost0 (V ∗,W ∗). Note also that

under this mapping

g · (C, i, j) = (gCg−1, gi, jg−1) 7→((gCg−1)∗, (jg−1)∗, (gi)∗)


= ((g∗)−1C∗g∗, (g∗)−1j∗, i∗g∗)

= (g∗)−1 · (C∗, j∗, i∗)

for all g ∈ GV and (C, i, j) ∈Mst0 (V,W ). Hence, we have a commutative diagram

GV ∗ ×Mcost0 (V ∗,W ∗) Mcost

0 (V ∗,W ∗)

GV ×Mst0 (V,W ) Mst

0 (V,W )

,

where the horizontal arrows represent the group action and the mapping GV ×

Mst0 (V,W )→ GV ∗ ×Mcost

0 (V ∗,W ∗) is given by (g, (C, i, j)) 7→ ((g∗)−1, (C∗, j∗, i∗)).

Corollary 3.12. The varieties M(V,W ) and M∗(V ∗,W ∗) are isomorphic (as vari-

eties).

Proof: This follows from Lemma 3.11 and Lemma 2.3.

Corollary 3.13. The varietes Mst0 and M are smooth.

Proof: By Corollary [21, Lemma 3.10, Corollary 3.12], we have that Mcost0 and

M∗ are smooth varieties. Thus, by Lemma 3.11 and Corollary 3.12, Mst0 and M are

also smooth varieties.

Let (C, i, j) ∈Mst0 . The tangent space of Mst

0 at (C, i, j) is

T(C,i,j)(Mst0 ) = T(C,i,j)(µ

−1(0)) = ker dµ,


and one can easily check that the differential of the moment map µ at (C, i, j) is

dµ : M→ LV,V ,

(D, a, b) 7→ ε(C)D + ε(D)C + ib+ aj.

By Corollary 2.6, the tangent space T[C,i,j](M) can then be identified with the quo-

tient ker dµ/ im dσ(C,i,j). Here, σ(C,i,j) : GV → Mst0 is the map given by g 7→

(gCg−1, gi, jg−1), and so

dσ(C,i,j) : LV,V → T(C,i,j)(Mst0 ) = ker dµ,

X 7→ (XC − CX,Xi,−jX).

Lemma 3.14. Let (C, i, j) ∈Mst0 . Then

1. dσ(C,i,j) is injective and dµ (at (C, i, j)) is surjective, and

2. the tangent space of M at [C, i, j] may be identified with the middle cohomology

of the following complex:

LV,Vdσ(C,i,j)

−−−−→ EV ⊕ LW,V ⊕ LV,Wdµ−−− LV,V . (3.4)

Proof: The proof of (1) is completely analogous to [15, Lemma 3.2]. Part (2), as

above, simply follows from Corollary 2.6.

It is worth noting that, up to isomorphism, the varieties Mst0 (V,W ) and M(V,W )

are parametrized by the graded dimensions of V and W . Indeed, let v = (vk),w =

(wk) ∈ NQ0 and, for i = 1, 2, let V i, W i be Q0-graded vector spaces such that

dimV ik = vk and dimW i

k = wk for all k ∈ Q0. Fix Q0-graded vector space isomor-

phisms f : V 1 → V 2 and h : W 1 → W 2. Then we define maps

Mst0 (V 1,W 1)→Mst

0 (V 2,W 2),


(C, i, j) 7→ (fCf−1, fih−1, hif−1),

and

GV 1 → GV 2 ,

g 7→ fgf−1.

One can easily check that these maps are isomorphisms of varieties and groups, re-

spectively. Thus, Mst0 (V 1,W 1) ∼= Mst

0 (V 2,W 2) and, by Lemma 2.3, M(V 1,W 1) ∼=

M(V 2,W 2). Hence, we define one standard representative for each isomorphism class:

Mst0 (v,w) := Mst

0

⊕k∈Q0

Cvk ,⊕k∈Q0

Cwk

,

and

M(v,w) := M

⊕k∈Q0

Cvk ,⊕k∈Q0

Cwk

= Mst0 (v,w)/Gv,

where Gv =∏

k∈Q0GLvk(C).

From now on, we restrict ourselves to the case where Q is a quiver of type Am−1,

m ∈ N+ (with the case m = 1 corresponding to the quiver consisting of one vertex

and one loop). We label the vertices of Q by 0, 1, . . . ,m − 1, and hence we can

identify Q0 = Q0 with Zm. In this case, we have that Q is the quiver:

1 2 · · · m− 1

0

We denote by M(m; v,w) (resp. M(m; v,w)) the variety M(v,w) (resp. M(v,w))

corresponding to a quiver of type Am−1 (recall that these varieties do not depend on

the orientation ofQ, thus M(m; v,w) and M(m; v,w) are well-defined). Of particular

interest to us will be the case m = 1, for which we introduce the special notation:

M(s, n) := M(1; (n), (s)), M(s, n) := M(1; (n), (s)).


Note that in this case

M(s, n) = Hom(Cn,Cn)⊕ Hom(Cn,Cn)⊕ Hom(Cs,Cn)⊕ Hom(Cn,Cs),

and so we will denote elements of M(s, n) by (A,B, i, j) and the elements ofM(s, n)

by [A,B, i, j], where, by convention, A represents the linear map associated to the

loop in Q1 and B the linear map associated to the loop in Q1. The moment map µ

then simplifies to

µ(A,B, i, j) = [A,B] + ij.

Thus, by [22, Theorem 2.1], M(s, n) is isomorphic to the moduli space of framed

torsion-free sheaves on P2 with rank s and second Chern class c2 = n.

Fix an (s+ 1)-dimensional torus

T = (C∗)s × C∗,

and denote elements of T by (e, t), where e = (e1, . . . , es) ∈ (C∗)s and t ∈ C∗. We

have a natural action of (C∗)s on Cs given by

e · (w1, . . . , ws) = (e1w1, . . . , esws),

for all e ∈ (C∗)s and (w1, . . . , ws) ∈ Cs. For each c ∈ Zs, we have an action of T on

M(s, n) given by

(e, t) · (A,B, i, j) = (tA, t−1B, ie−1t−c, etcj),

where

tc := (tc1 , . . . , tcs).

The torus action preserves the space M st0 (s, n) and commutes with the action of

GLn(C), thus we have a well-defined action on M(s, n):

(e, t) · [A,B, i, j] = [tA, t−1B, ie−1t−c, etcj]. (3.5)


LetMc(s, n) denote the moduli spaceM(s, n) with the torus action given by Equa-

tion (3.5).

We restrict our focus for now to the case s = 1. It is known that (A,B, i, j) ∈

M st0 (1, n) =⇒ j = 0 (see [22, Proposition 2.7]), thus we will simply write (A,B, i)

for an element (A,B, i, 0) ∈ M st0 (1, n). Let ω := e2π

√−1/m. The finite cyclic group of

order m acts on Mc(1, n) via the embedding

Zm → T,

k 7→ (1, ωk).(3.6)

That is

k · [A,B, i] = [ωkA, ω−kB,ω−kci],

for all k ∈ Zm and [A,B, i] ∈Mc(1, n). Recall that the fixed point setMc(1, n)Zm is

a closed subvariety of Mc(1, n) (see the discussion preceding Lemma 2.9). The goal

for the remainder of this chapter will be to describe Mc(1, n)Zm in terms of quiver

varieties of type Am−1.

Remark 3.15. Let

Fc(n) :=

(A,B, i) ∈M st0 (1, n) | [A,B, i] ∈Mc(1, n)Zm

,

i.e. we let Fc(n) be the preimage of Mc(1, n)Zm under the projection M st0 (1, n)

Mc(1, n). Thus, Fc(n) is a closed subvariety of M st0 (1, n). We have that (A,B, i) ∈

Fc(n) if and only if there exists a g ∈ GLn(C) such that

ωA = g−1Ag,

ω−1B = g−1Bg,

ω−ci = g−1i.

(3.7)

Note that since GLn(C) acts freely on M st0 , such a g, if it exists, is unique. Assume

that (A,B, i) ∈ Fc(n). By Remark 3.3, every v ∈ V may be written as a finite sum

v =∑p,q

λpqApBqi,


where λpq ∈ C (note that A and B commute since the moment map implies that

[A,B] = 0) and we identify the map i : C→ Cn with i(1). For any p and q,

g(λpqApBqi) = ω−pλpqA

pgBqi = ωq−pλpqApBqgi = ωq−p+cλpqA

pBqi.

Thus, we have a weight space decomposition

Cn =⊕k∈Zm

Vk(A,B, i), Vk(A,B, i) := v ∈ Cn | gv = ωkv.

Note that by the above calculation, it is easily seen that

Vk(A,B, i) = spanApBqi | (q − p) ≡ (k − c) mod m. (3.8)

We observe that i ∈ Vc (where c is the equivalency class of c in Zm) and the restrictions

of A and B to Vk yield maps

A|Vk(A,B,i) : Vk(A,B, i)→ Vk−1(A,B, i) and B|Vk(A,B,i) : Vk(A,B, i)→ Vk+1(A,B, i).

Conversely, suppose we have a weight space decomposition Cn =⊕

k∈Zm Vk such that

A|Vk : Vk → Vk−1 and B|Vk : Vk → Vk+1 and i ∈ Vc. Set

g =∏k∈Zm

ωk idVk .

Then g−1Ag = ωA, g−1Bg = ω−1B and g−1i = ω−ci, and so (A,B, i) ∈ Fc(n).

Moreover, Vk(A,B, i) = Vk for each k ∈ Zm.

Lemma 3.16. Let (A,B, i) ∈ Fc(n) and h ∈ GLn(C). Then

Vk(h · (A,B, i)) = hVk(A,B, i),

for all k ∈ Zm.

Proof: Write

(A′, B′, i′) = h · (A,B, i) = (hAh−1, hBh−1, hi).


Let g ∈ GLn(C) be the (unique) element satisfying (3.7) for (A,B, i). Let g′ = hgh−1.

Then

(g′)−1A′g′ = (hg−1h−1)(hAh−1)(hg−1h−1) = h(g−1Ag)h−1 = ω(hAh−1) = ωA′.

Similarly, (g′)−1B′g′ = ω−1B′. Moreover,

(g′)−1i′ = (hg−1h−1)(hi) = h(g−1i) = ω−chi = ω−ci′.

Hence, g′ is the unique element in GLn(C) satisfying (3.7) for (A′, B′, i′). Thus,

Vk(A,B, i) = v ∈ Cn | gv = ωkv, and Vk(A′, B′, i′) = v ∈ Cn | g′v = ωkv.

For each v ∈ Vk(A,B, i),

g′(hv) = (hgh−1)(hv) = h(gv) = ωk(hv),

and so hv ∈ Vk(A′, B′, i′). Thus, h is a linear map Vk(A,B, i) → Vk(A′, B′, i′). Since

h is invertible, Vk(A′, B′, i′) = hVk(A,B, i).

Lemma 3.17. 1. The variety Mc(1, n)Zm is smooth.

2. The tangent space of Mc(1, n)Zm at [A,B, i] may be identified with the middle

cohomology of the following complex:

Ldσ(A,B,i)

−−−−−→ E− ⊕ E+ ⊕ Hom(C, Vc)⊕ Hom(Vc,C)dµ−−− L, (3.9)

where L =⊕

k∈Zm Hom(Vk, Vk) and E± =⊕

k∈Zm Hom(Vk, Vk±1).

Proof: For Part (1), note that Zm is a compact Lie group, and thus, the category

of finite-dimensional linear representations of Zm is semisimple. Therefore, by [9,

Proposition 1.3], Mc(1, n)Zm is smooth. For Part (2), we first note that the tangent


space of Mc(1, n) is obtained by setting V = Cn and W = C in Complex (3.4), i.e.

the tangent space of Mc(1, n) is the middle cohomology of

Hom(Cn,Cn)dσ(A,B,i)

−−−−−→Hom(Cn,Cn)⊕ Hom(Cn,Cn)

⊕

Hom(C,Cn)⊕ Hom(Cn,C)

dµ−−− Hom(Cn,Cn). (3.10)

Therefore, by Lemma 2.9, the tangent space ofMc(1, n)Zm ∼= (ker dµ/ im dσ(A,B,i))Zm .

Let Zm act on Cn by

ω · v = gv,

for all v ∈ V , where g is the element in GLn(C) satisfying properties (3.7) for (A,B, i).

Likewise, let Zm act on C by

ω · u = ωcu,

for all u ∈ C. Then Hom(Cn,Cn), Hom(C,Cn) and Hom(Cn,C) naturally become

Zm-modules. In order for dσ(A,B,i) and dµ to be Zm-module maps, we introduce the

one-dimensional modules ω and ω−1, (on which ω acts by multiplication by ω and

ω−1, respectively), and modify Complex (3.10) to

Hom(Cn,Cn)dσ(A,B,i)

−−−−−→ωHom(Cn,Cn)⊕ ω−1 Hom(Cn,Cn)

⊕

Hom(C,Cn)⊕ Hom(Cn,C)

dµ−−− Hom(Cn,Cn),

where we use juxtaposition to indicate the tensor product. One checks that the

action of Zm on ker dµ/ im dσ(A,B,i) induced by the action on Cn and C matches

the action described in Remark 2.8. Thus, by Lemma 2.10, (ker dµ/ im dσ(A,B,i))Zm ∼=

ker dµZm/ im dσ(A,B,i)Zm , where dµZm and dσ

(A,B,i)Zm are the restrictions of dµ and dσ(A,B,i)

to the fixed points of their respective domains. Now, f ∈ Hom(Cn,Cn)Zm if and only

if f = gfg−1, which occurs if and only if f |Vk : Vk → Vk. Therefore,

Hom(Cn,Cn)Zm ∼=⊕k∈Zm

Hom(Vk, Vk).


Similarly, one can show that

(ω±1 Hom(Cn,Cn))Zm ∼=⊕k∈Zm

Hom(Vk, Vk∓1), Hom(C,Cn)Zm ∼= Hom(C, Vc),

and Hom(Cn,C)Zm ∼= Hom(Vc,C),

which completes the proof.

Let (C, i, j) ∈ M(m; v,1c). We identify⊕

k∈Zm Cvk with C|v| by identifying 1k`

with 1v0+···+vk−1+`, where 1k`vk`=1 and 1`|v|`=1 denote the standard bases of Cvk and

C|v|, respectively. Let AC , BC ∈ Hom(C|v|,C|v|

)be the maps determined by

(AC)|Cvk = ε(ρ)Cρ and (BC)|Cvk = Cτ , (3.11)

for each k ∈ Zm, ρ : k → k− 1 in Q1 and τ : k → k+ 1 in Q1 (note that in the m = 2

case, we simply make a choice as to which arrow of Q1 corresponds to k → k+ 1 and

which one corresponds to k → k − 1). We thus have a mapping

M(m; v,1c)→M(1, |v|),

(C, i, j) 7→ (AC , BC , i, j).(3.12)

Suppose that µ(C, i, j) = 0. For every vertex k ∈ Q0, we have four arrows incident

to k as in the following diagram.

k − 1 k k + 1α

α

β

β(3.13)

Thus, the Hom(Cvk ,Cvk) component of ε(C)C is

(ε(C)C)k = ε(α)CαCα + ε(β)CβCβ = −Cα(ε(α)Cα) + (ε(β)Cβ)Cβ = [AC , BC ]|Cvk .

Hence, ε(C)C = [AC , BC ]. Therefore,

[AC , BC ] + ij = ε(C)C + ij = µ(C, i, j) = 0.


Now, suppose (C, i, j) is stable. Furthermore, suppose S is a subspace of C|v| such

that AC(S) ⊆ S, BC(S) ⊆ S and im(i) ⊆ S. For each k ∈ Zm, let Sk be the vector

space spanned by elements of the form:

D`1D`2 · · ·D`pi(1),

where each ì ∈ 1, 2, p ∈ N, D1 = AC , D2 = BC and

#ì | ì = 2 −#ì | ì = 1 ≡ k − c mod m.

Then S =⊕

k∈Zm Sk and, by construction, S is thus a C-invariant subspace of C|v|.

Since (C, i, j) is stable, S = C|v|. Therefore, (AC , BC , i, j) is stable. In particular, this

means that if (C, i, j) ∈Mst0 (m; v,1c), then (AC , BC , i, j) ∈M st

0 (1, |v|). In particular,

by [22, Proposition 2.7], this implies j = 0. Since this fact turns out to be rather

important, we highlight it with the following lemma.

Lemma 3.18. If (C, i, j) ∈Mst0 (m; v,w) with |w| = 1, then j = 0.

By the discussion above, the mapping (3.12) maps Mst0 →M st

0 . Thus, by Equa-

tion (3.11), Lemma 3.18 and Remark 3.15, we may define a morphism of varieties

ϕv : Mst0 (m; v,1c)→ Fc(|v|),

(C, i, 0) 7→ (AC , BC , i).

Lemma 3.19. The map ϕv induces a morphism of varieties

ϕv : M(m; v,1c)→Mc(1, |v|)Zm ,

[C, i, 0] 7→ [AC , BC , i].

In particular, one has a morphism of varieties ϕ :∐|v|=nM(m; v,1c)→Mc(1, n)Zm,

given by ϕ =∐|v|=n ϕv.


Proof: Let ψ : Gv → GL|v|(C) be the group homomorphism induced by our

identification of⊕

k∈Zm Cvk with C|v|, i.e. partitioning GL|v|(C) into m2 blocks of size

vi × vj, i, j = 0, 1, . . . ,m − 1, we embed GLvk(C) into the (k, k)-th diagonal block

of GL|v|(C), for each k = 0, 1, . . . ,m − 1. One then has the following commutative

diagram:

Gv ×Mst0 (m; v,1c) Mst

0 (m; v,1c)

GL|v|(C)× Fc(|v|) Fc(|v|),

ψ × ϕ ϕ

where the horizontal arrows represent the group action. The result then follows by

Lemma 2.3.

Theorem 3.20. The map ϕ from Lemma 3.19 is an isomorphism. Therefore,

Mc(1, n)Zm ∼=∐|v|=n

M(m; v,1c),

as varieties.

Proof: We first begin by showing that ϕ is bijective. Suppose that (C, i, 0) ∈

Mst0 (m; v,1c) and (D, a, 0) ∈Mst

0 (m; u,1c), with |v| = |u| = n, are such that

ϕ[C, i, 0] = ϕ[D, a, 0],

i.e. [AC , BC , i] = [AD, BD, a]. Then there exists g ∈ GLn(C) such that (AC , BC , i) =

g · (AD, BD, a). For each k ∈ Zm,

Cvk = Vk(AC , BC , i) = g(Vk(AD, BD, a)) = g(Cuk),

where the second equality follows from Lemma 3.16. In particular, v = u. Moreover,

since g|Cvk : Cvk → Cvk , we may view g as an element of Gv. One then easily verifies

that (C, i, 0) = g · (D, a, 0). Thus, [C, i, 0] = [D, a, 0], and so ϕ is injective.


Now, let (A,B, i) ∈ Fc(n). Set vk = dimVk(A,B, i) and choose a graded linear

isomorphism

f :⊕k∈Zm

Vk(A,B, i)→⊕k∈Zm

Cvk .

Then [A,B, i] = [fAf−1, fBf−1, fi]. Define (C, a, 0) ∈M (m; v,1c) by

Cρ =

ε(ρ)(fAf−1)|Cvk , if ρ : k → k − 1,

(fBf−1)|Cvk , if ρ : k → k + 1,

and a = fi,

for all ρ ∈ Q1. We claim that (C, a, 0) ∈Mst0 (m; v,1c). To show that µ(C, a, 0) = 0,

recall that for each k ∈ Zm, one has 4 arrows incident to k as in Diagram (3.13).

Thus, the Hom(Cvk ,Cvk)-component of ε(C)C is

(ε(C)C)k = ε(α)CαCα + ε(β)CβCβ

= ε(α)ε(α)(fBf−1)|Cvk−1 (fAf−1)|Cvk + ε(β)2(fAf−1)|Cvk+1 (fBf−1)|Cvk

= [fAf−1, fBf−1]|Cvk .

Thus, applying the moment map to (C, a, 0),

µ(C, a, 0) = ε(C)C = [fAf−1, fBf−1] = f [A,B]f−1 = 0,

where the last equality follows from the fact that µ(A,B, i) = 0. Therefore, (C, a, 0) ∈

µ−1(0). To show that (C, a, 0) is stable, suppose that S =⊕

k∈Zm Sk is a C-invariant

subspace of⊕

k∈Zm Cvk with a ∈ S. Then by construction, A(f−1(S)), B(f−1(S)) ⊆

f−1(S) and i ∈ f−1(S). Since (A,B, i) is stable, f−1(S) =⊕

k∈Zm Vk, and thus

S =⊕

k∈Zm Cvk . Hence, (C, a, 0) is stable. Therefore, (C, a, 0) ∈Mst0 (m; v,1c) and

ϕ[C, a, 0] = [fAf−1, fBf−1, fi] = [A,B, i].

Hence, ϕ is surjective.

Next, we show that ϕ is an etale morphism. Recall that we identify the tangent

spaces of M andMZm with the middle cohomologies of Complex (3.4) and Complex


(3.9), respectively. By construction of ϕ, the induced map dϕ on the tangent spaces

is

dϕ : T[C,i,0](M)→ T[AC ,BC ,i](MZm),

(D, a, b) + im dσ(C,i,j) 7→ (D−, D+, a, b) + im dσ(AC ,BC ,i),

where (D−)k = ε(ρ)Dρ:k→k−1 and (D+)k = Dρ:k→k+1 for all k ∈ Zm. Clearly, dϕ is

injective. Moreover, by Lemma 3.14 and Lemma 3.17, T[C,i,0](M) and T[AC ,BC ,i](MZm)

have the same dimension, and hence dϕ is an isomorphism. By Lemma 2.11, ϕ is an

isomorphism.

Chapter 4

Vector Bundles and Geometric

Operators

In this chapter, we describe how to obtain so-called “geometric operators” on the

(localized) equivariant cohomology of smooth algebraic varieties; this method was

first introduced in [3] . The methods outlined in this chapter will serve as our main

tool for constructing our geometric versions of the Clifford, Heisenberg and Chevalley

operators in the next chapter. We do not review equivarient cohomology theory here,

but instead refer the reader to such expository papers as [27] or [2]. Our constructions

rely heavily on the Localization Theorem (see Theorem 4.1). The reader may wish

to consult [1, Appendix to Chapter 6] for more information on localization.

Let G = (C∗)d be a d-dimensional torus and, for each j = 1, . . . , d, we denote

the 1-dimensional G-module

(g1, . . . , gd) 7→ gj,

by gj. Let pt denote the space consisting of a single point equipped with the trivial

action of the torus G. Let tj denote the first Chern class of

gj → pt,

63

4. Vector Bundles and Geometric Operators 64

for each j = 1, . . . , d. Note that the tj are elements of degree 2. Recall that the

equivariant cohomology of pt is

H∗G(pt) = C[t1, . . . , td].

Let X be a space with a G-action. Then H∗G(X) is an H∗G(pt)-module. We consider

the localized equivariant cohomology of X:

H∗G(X) := H∗G(X)⊗C[t1,...,td] C(t1, . . . , td).

Unless otherwise noted, “cohomology” will always mean “localized equivariant coho-

mology”. Let

i : XG → X,

be the inclusion of the G-fixed points and let

p : XG pt .

The advantage of localized equivariant cohomology over nonlocalized equivariant co-

homology is that its study can be reduced to the cohomology of the G-fixed points.

We will only be interested in the case where X is a smooth variety with finitely many

G-fixed points. In this situation, we have the following theorem.

Theorem 4.1 (Localization Theorem). The following map is an isomorphism of

algebras:

H∗G(X)→ H∗G(XG) =⊕x∈XG

H∗G(pt),

α 7→(i∗x(α)

eG(Tx)

)x∈XG

,

(4.1)

where ix : x → X, Tx is the tangent space of x in X, and eG(Tx) is the equivariant

Euler class of Tx. The inverse of (4.1) is given by the Gysin map i∗ : H∗G(XG) →

H∗G(X).


Proof: This is a restatement of [5, Proposition 9.1.2] in the case that X has

finitely many fixed points.

Suppose now that X has real dimension 4n, for some n ∈ N. We define a bilinear

form 〈-, -〉X on the middle degree localized equivariant cohomology H2nG (X) by

〈a, b〉X := (−1)np∗(i∗)−1(a ∪ b), (4.2)

where i∗ is invertible by the Localization Theorem. We can extend this idea to a

product of varieties. Indeed, suppose X1, X2 are varieties of real dimension 4n1 and

4n2, respectively. We define a bilinear form 〈-, -〉X1×X2 on H2(n1+n2)G (X1 ×X2) by

〈a, b〉 := (−1)n2p∗((i1 × i2)∗)−1(a ∪ b), (4.3)

where i1 and i2 are the inclusions of the G-fixed points into the first and second

factors, respectively. An element α ∈ H2(n1+n2)G (X1 ×X2) then defines an operator

α : H2n1G (X1)→ H2n2

G (X2), (4.4)

by using the bilinear form to define structure constants:

〈α(x), y〉X2 := 〈α, x⊗ y〉X1×X2 .

Thus, an element α ∈ H2(n1+n2)G (X1 ×X2) will be called a geometric operator.

Recall that the torus T = (C∗)s × C∗ acts on Mc(s, n) via

(e, t) · [A,B, i, j] = [tA, t−1B, ie−1t−c, etcj], (4.5)

for all (e, t) ∈ T and [A,B, i, j] ∈ Mc(s, n). Let T• = C∗ and define an action of T•

on Mc(s, n) via the embedding

T• → T

z 7→ (1, z, z2, . . . , zs−1, 1).


Similar to Remark 3.15, one sees that [A,B, i, j] ∈ Mc(s, n)T• if and only if there

exists a group homomorphism g : T• → GLn(C) such that

g(z)−1Ag(z) = A,

g(z)−1Bg(z) = B,

g(z)−1i = i(1, z−1, . . . , z1−s),

jg(z) = (1, z, . . . , zs−1)j.

(4.6)

Define

V k = V k(A,B, i, j) := v ∈ Cn | g(z)v = zk−1v. (4.7)

By the stability of (A,B, i, j), we have that Cn =⊕s

k=1 Vk. Moreover,

A(V k), B(V k), i(C1k) ⊆ V k, and j(Vk) ⊆ C1k,

for all k = 1, . . . , s. Conversely, if there exists a decomposition Cn =⊕s

k=1 Uk such

that A(Uk), B(Uk), i(1k) ⊆ Uk and j(Uk) ⊆ C1k, then we may define a group

homomorphism g : T• → GLn(C) by defining g(z)|Uk = zk−1 idUk . One easily checks

that g satisfies the conditions of (4.6) and Uk = V k(A,B, i, j). Thus, [A,B, i, j] ∈

Mc(s, n)T• .

Now suppose n = (n1, . . . ,ns) ∈ Ns such that |n| = n. Define

Mc(n) :=Mc1(1,n1)× · · · ×Mcs(1,ns).

Identify⊕

k Cnk with Cn by identifying 1k` with 1n1+···+nk−1+`, where 1k`nk`=1 is the

standard basis of Cnk . An element ([A1, B1, i1], . . . , [As, Bs, is]) ∈Mc(n) then deter-

mines an element [A,B, i, 0] ∈Mc(s, n)T• by defining

A|Cnk := Ak, B|Cnk := Bk, i = i1 + · · ·+ is. (4.8)

One can then check that we have a well-defined map∐|n|=n

Mc(n)→Mc(s, n)T• ,

([A1, B1, i1], . . . , [As, Bs, is]) 7→ [A,B, i, 0],

(4.9)


where [A,B, i, 0] is defined as in (4.8).

Lemma 4.2. The map (4.9) is an isomorphism of varieties. Thus, Mc(s, n)T• ∼=∐|n|=nMc(n).

Proof: This is a straight-forward generalization of [23, Lemma 3.2].

We fix once and for all an r ∈ N+ and a partition of r of length s,

r := (r1, . . . , rs).

Let R′ = lcmr1, . . . , rs and define

R :=

R′, if R′

(1rk

+ 1r`

)∈ 2Z for all k, `,

2R′, otherwise.

Consider the product variety Mc(n) =Mc1(1,n1)× · · · ×Mcs(1,ns). Each compo-

nent Mc`(1,n`), for ` = 1, . . . , s, carries with it the action of a 2-dimensional torus

T = C∗ × C∗ (given by setting s = 1 in Equation (4.5)). Let T? = (C∗)s × C∗ and

define a T?-action on Mc`(1,n`) via the map

T? → T,

(e, t) 7→ (e`, tR/r`).

That is, T? acts on Mc`(1,n`) by

(e, t) ? [A`, B`, i`] = (e`, tR/r`) · [A`, B`, i`] = [tR/r`A`, t

−R/r`B, i`e−1` t−c`R/r` ],

for all (e, t) ∈ T? and [A`, B`, i`] ∈ Mc`(1,n`). Then T? acts on the product Mc(n)

by acting on each of its components, i.e.

(e, t) ? ([A1, B1, i1], . . . , [As, Bs, is]) = ((e, t) ? [A1, B1, i1], . . . , (e, t) ? [As, Bs, is]).


Lemma 4.3. The set Mc(n)T? is in one-to-one correspondence with the set

(I1, . . . , Is) | I` is a semi-infinite monomial of charge c` and energy n`.

Proof: We first prove that

Mc(n)T? =Mc1(1,n1)T × · · · ×Mcs(1,ns)T .

Let ([A1, B1, i1], . . . , [As, Bs, is]) ∈ Mc(n)T? . Fix ` ∈ 1, . . . , s. For all (e, t) ∈ T =

C∗ × C∗, choose ξ ∈ C∗ such that ξR/r` = t. Then

(e, t) · [A`, B`, i`] = ((1, . . . , e, . . . , 1), ξ) ? [A`, B`, i`] = [A`, B`, i`].

Therefore,Mc(n)T? ⊆Mc1(1,n1)T ×· · ·×Mcs(1,ns)T . The reverse inclusion follows

by construction of the action of T? on Mc(n).

Now, by [21, Proposition 2.9], Mc`(1,n`)T is in one-to-one correspondence with

the set of Young diagrams of size n`, which is itself in one-to-one correspondence with

the set of semi-infinite monomials of charge c` and energy n` by Lemma 1.9.

In light of Lemma 4.3, we will henceforth identify points ofMc(n)T? with s-tuples

of semi-infinite monomials.

Define an action of ZR on Mc(n) via the embedding

ZR → T?,

k 7→ (1, ωk),

where ω = e2π√−1/R. Then ZR acts on the `-th component,Mc`(1,n`), ofMc(n) via

the embedding

ZR → T,

k 7→ (1, ωR/r`) = (1, e2π√−1/r`).


Thus,

Mc(n)ZR =Mc1(1,n1)Zr1 × · · · ×Mcs(1,ns)Zrs ,

where the action of Zr` on Mc`(1,n`) is defined as in Equation (3.6). By Theorem

3.20, we know that Mc`(1,n`)Zr` ∼=

∐|v`|=n`

M(r`; v`,1c`). Hence, we define

Mc(v1, . . . ,vs) := M(r1; v1,1c1)× · · · ×M(rs; v

s,1cs),

and obtain

Mc(n)ZR ∼=∐|v`|=n`

Mc(v1, . . . ,vs).

We summarize the various fixed point varieties with the following diagram of inclu-

sions:

Mc(s, n) ⊇Mc(s, n)T• ∼=∐

nMc(n) ⊇∐

nMc(n)ZR ⊇∐

nMc(n)T? .

∼=∐v` Mc(v

1, . . . ,vs)

(4.10)

We now consider the localized T?-equivariant cohomology ofMc(n). Denote the

one-dimensional T?-modules

(e, t) 7→ ek, and (e, t) 7→ t,

by ek and t, respectively, and denote the tensor product of such modules by juxtapo-

sition. Moreover, we denote the first Chern classes of

ek 7→ pt, and t 7→ pt,

by bk and ε, respectively. Thus,

H∗T?(Mc(n)) = HT?(Mc(n))⊗C[b1,...,bs,ε] C(b1, . . . , bs, ε).


SinceMc(n) has real dimension 4|n|, we define a bilinear form 〈-, -〉n,c onH2|n|T?

(Mc(n))

as in (4.2), induced by

i :Mc(s, n)T? →Mc(n) and p :Mc(n)T? pt .

We extend to a bilinear form 〈-, -〉 on⊕

n,cH2|n|T?

(Mc(n)) ∼=⊕

n,cH2nT?

(Mc(s, n)T•

)by

〈-, -〉 :=∑n,c

〈-, -〉n,c.

One also defines a bilinear form 〈-, -〉n,m,c,d on H2(|n|+|m|)T?

(Mc(n) ×Md(m)) as in

Equation (4.3), which we extend to a bilinear form 〈-, -〉 on⊕

n,m,c,dH2(|n|+|m|)T?

(Mc(n)×

Md(m)) by

〈-, -〉 :=∑

n,m,c,d

〈-, -〉n,m,c,d.

Our goal will be to construct geometric operators on⊕

n,cH2|n|T?

(Mc(n)) as in Equa-

tion (4.4). In order to simplify computations, it will be useful for us to introduce

an orthonormal C(b1, . . . , bs, ε)-basis for H∗T?(Mc(n)). For each I ∈ Mc(n)T? , the

T?-action on Mc(n) induces an action on the tangent space TI = TI(Mc(n)). The

decomposition of TI into one-dimensional T? modules is given in the following lemma.

Lemma 4.4. Let I = (I1, . . . , Is) ∈Mc(n)T?. Then, as a T?-module,

TI ∼=s⊕`=1

⊕(i,j)∈λ(I`)

(t−hλ(I`)(i,j)R/r` ⊕ thλ(I`)(i,j)R/r`)

,

where λ(I`) is the Young diagram associated to I` and hλ(I) is the relative hook length

(see equations (1.2) and (1.1)).

Proof: We have that

TI(Mc(n)) ∼=s⊕`=1

TI`(Mc`(1,n`)).


The tangent space of Mc`(1,n`) at I` may then be computed by replacing t by tR/r`

in [15, Proposition 2.2].

It will be convenient to use the decomposition TI = T +I ⊕ T

−I , where

T ±I :=s⊕`=1

⊕(i,j)∈λ(I`)

t±hλ(I`)(i,j)R/r`

.

Lemma 4.5. For I ∈Mc(n)T?, the equivariant Euler classes of T +I and T −I are given

by

eT?(T +I ) =

s∏`=1

∏(i,j)∈λ(I`)

hλ(I`)(i, j)R

r`ε

,

eT?(T −I ) =s∏`=1

∏(i,j)∈λ(I`)

−hλ(I`)(i, j)R

r`ε

= (−1)|n|eT?(T +I ).

Proof: This follows directly from the definitions of T +I and T −I .

For each I ∈Mc(n)T? , let

[I] :=i∗(1I)

eT?(T −I )∈ H2|n|

T?(Mc(n)),

where 1I is the unit in H∗T?(pt) and eT?(T −I ) is to be interpreted as an invertible ele-

ment in this ring. Since the elements 1I form a C(b1, . . . , bs, ε)-basis of the cohomlogy⊕n,cH∗T?

(Mc(n)T?

), by the Localization Theorem (Theorem 4.1), the elements [I]

form a C(b1, . . . , bs, ε)-basis of⊕

n,cH∗T?(Mc(n)).

Lemma 4.6. The [I] are orthonormal with respect to the bilinear form 〈-, -〉.

Proof: The proof is completely analogous to [15, Proposition 2.4].


Define

A = spanC

[I] | I ∈Mc(n)T? , n ∈ Ns, c ∈ Zs. (4.11)

Then A is a full C-lattice in⊕

n,cH∗T?(Mc(n)). It will be useful for us to consider

the following gradings on A:

A =⊕n,c

Ac(n), Ac(n) :=

[I] | I ∈Mc(n)T?,

and

A =⊕c∈Z

A(c), A(c) :=

[I] | I ∈Mc(n)T? s.t. |c| = c.

Corollary 4.7. The restriction of 〈-, -〉 to A is non-degenerate and C-valued.

Proof: This follows directly from Lemma 4.6.

Remark 4.8. Notice that via the Kunneth formula

H∗T?(Mc(n)) ∼= H∗T?(Mc1(1,n1))⊗ · · · ⊗ H∗T?(Mcs(1,ns)),

the element [I] ∈ H∗T?(Mc(n)), where I = (I1, . . . , Is), maps to

[I1]⊗ · · · ⊗ [Is] ∈ H∗T?(Mc1(1,n1))⊗ · · · ⊗ H∗T?(Mcs(1,ns)).

Therefore,

Ac(n) ∼= Ac1(n1)⊗ · · · ⊗Acs(ns).

Let X be an algebraic variety with a T?-action, and let E → X be an T?-

equivariant vector bundle. We will denote the k-th equivariant Chern class of E by

ck(E). Note that ck(E) ∈ H2kT?

(X). The following lemma will act as our main tool in

constructing geometric operators in the following chapter.


Lemma 4.9. Let I ∈ Mc(n)T? and J ∈ Md(m)T?, let E be an equivariant vector

bundle on Mc(n)×Md(m) and let β ∈ H2kT?

(Mc(n)×Md(m)). Then

〈β ∪ c|n|+|m|−k(E)[I], [J]〉 =βI,J ∪ c|n|+|m|−k(E(I,J))

eT?(T −I )eT?(T +J )

,

where c|n|+|m|−k(E(I,J)) ∈ H∗T?(pt) = C[b1, . . . , bs, ε] is the polynomial given by the

equivariant Chern class of the fiber E over the point (I,J) and βI,J = i∗I,J(β), where

iI,J : (I,J) →Mc(n)×Md(m) is the inclusion of the fixed point.

Proof: See [15, Lemma 2.6].

Remark 4.10. The precise statement of [15, Lemma 2.6] differs slightly from ours

(since we use the variety Mc(n) under the action of T? rather than Mc(s, n) under

the action of T ). However, every step in the proof of [15, Lemma 2.6] applies to

Lemma 4.9; thus the two lemmas are essentially the same.

The next step will be to construct T?-equivariant vector bundles over Mc(n) ×

Md(m) whose Chern classes will define the appropriate geometric operators (as our

ultimate goal is to construct geometric versions of the Heisenberg, Clifford and Cheval-

ley operators from Chapter 1). We begin by defining vector bundles

Cn ×GLn(C) Mst0 (s, n)→Mc(s, n), and Cs ×Mc(s, n)→Mc(s, n),

which we denote by V = V(c, s, n) and W = W(c, s, n), respectively. Note that V

and W are simply the associated bundles of the trivial GLn(C)-bundles

Cn ×M st0 (s, n)→M st

0 (s, n), and Cs ×M st0 (s, n)→M st

0 (s, n).

The bundles V andW are T -equivariant with respect to the trivial action of T on Cn

and the natural action of T on Cs, respectively. Consider the Hom-bundle Hom(V ,V)


on Mc(s, n). We can define a global section s : Mc(s, n) → Hom(V ,V) by defining

s[A,B, i, j] to be the (well-defined) linear map

Cn ×GLn(C) [A,B, i, j]→ Cn ×GLn(C) [A,B, i, j],

GLn(C) · (v, (A,B, i, j)) 7→ GLn(C) · (Av, (A,B, i, j)).

By a slight abuse of notation, we will denote the section s by A. We similarly define

sections B, i and j of Hom(V ,V), Hom(W ,V) and Hom(V ,W), respectively.

One can extend this construction to a product of moduli spaces. The bun-

dle V(c, s, n) → Mc(s, n) may be extended to a vector bundle over the product

Mc(s, n)×Md(s,m) by:

V(c, s, n)×Md(s,m)→Mc(s, n)×Md(s,m).

We denote this vector bundle by V1 = V1(c,d, s, n,m). Likewise, we extend the bun-

dles W(c, s, n) → Mc(s, n), V(d, s,m) → Md(s,m) and W(d, s,m) → Md(s,m)

to bundles W1 = W1(c,d, s, n,m), V2 = V2(c,d, s, n,m) and W2 = W2(c,d, s, n,m)

over Mc(s, n) ×Md(s,m). We then have bundles Hom(Vk,Vk), Hom(Wk,Vk) and

Hom(Vk,Wk) as vector bundles over Mc(s, n) ×Md(s,m) with sections Ak, Bk, ik

and jk, where k = 1, 2. We define a three-term, T -equivariant complex of vector

bundles on Mc(s, n)×Md(s,m) by

Hom(V1,V2)ζ−→tHom(V1,V2)⊕ t−1 Hom(V1,V2)

⊕

Hom(W1,V2)⊕ Hom(V1,W2)

τ−→ Hom(V1,V2), (4.12)

where

ζ(X) =

XA1 − A2X

XB1 −B2X

Xi1

−j2X

, and τ

C

D

a

b

= A2D −DA1 + CB1 −B2C + i2b+ aj1.


Note that here the modules t±1 are the one-dimensional T -modules (e, t) 7→ t±1. One

verifies that τ ζ = 0.

Remark 4.11. We explain the presence of the “t” and “t−1” terms in Complex

(4.12); they are the result of the action of T on the section Ak and Bk. Let ζ1 :

Hom(V1,V2)→ Hom(V1,V2) be the first component of ζ. Denote the section “A1” by

s. To simplify notation, for every (I1, I2) ∈M st0 (s, n)×M st

0 (s,m), we will denote the

corresponding pair of orbits in Mc(s, n) ×Md(s,m) by (I1, I2). Moreover, we will

write s(I1,I2) := s(I1, I2) and v(I1,I2) := (GLn(C) · (v, I1), I2) for elements in the fiber

of V1 over the point (I1, I2). We wish to compute the action of T on s. For every

(e, t) ∈ T and (I1, I2) ∈M st0 (s, n)×M st

0 (s,m), with I1 = (A1, B1, i1, j1), the map

(e, t) · s(I1,I2) : (Cn ×GLn(C) I1)× I2 → (Cn ×GLn(C) I1)× I2,

is given by

((e, t) · s(I1,I2))(v(I1,I2)) = (e, t) · s(e,t)−1·(I1,I2)((e, t)−1 · v(I1,I2))

= (e, t) · s(e,t)−1·(I1,I2)(v(e,t)−1·(I1,I2))

= (e, t) · t−1Av(e,t)−1·(I1,I2) = t−1Av(I1,I2),

for all v ∈ Cn. Thus, the action of T on A1 (viewed as a section) is given by

(e, t) · A1 = t−1A1. Similarly, (e, t) · A2 = t−1A2. Therefore,

ζ1((e, t) ·X) = ((e, t) ·X)A1 − A2((e, t) ·X)

= (e, t) · (X((e, t)−1 · A1)− ((e, t)−1 · A2)X)

= (e, t) · (t(XA1 − A2X)) = t(e, t) · ζ1(X),

for all X ∈ Hom(V1,V2) and (e, t) ∈ T . Hence, ζ1 : Hom(V1,V2) → tHom(V1,V2) is

T -equivariant, thus justifying the extra factor “t” in Complex (4.12). One can show,

by a similar argument, that ζ and τ in Complex (4.12) are both T -equivariant.


Lemma 4.12. The cohomology ker τ/ im ζ of Complex (4.12) is a vector bundle on

Mc(s, n)×Md(s,m).

Proof: See [15, Lemma 3.2].

For n,m ∈ N, we will denote the vector bundle

ker τ/ im ζ →Mc(s, n)×Md(s,m),

by Kc,d(s, n,m). Notice that, by construction, one has the following vector bundle

on Mc(s, n)T• ×Md(s, n)T• :

(ker τ/ im ζ)T• →Mc(s, n)T• ×Md(s, n)T• ,

which we denote by Kc,d(s, n,m)T• . By Lemma 4.2,

Mc(s, n)T• ×Md(s,m)T• ∼=∐

|n|=n, |m|=m

Mc(n)×Md(m),

and so we may identify these varieties and consider the restriction of Kc,d(s, n,m)T• :

Kc,d(n,m) := Kc,d(s, n,m)T•|Mc(n)×Md(m),

which is a vector bundle on Mc(n) ×Md(m). On Mc`(1,n`) ×Md`(1,m`), the

product of the `-th components of Mc(n) and Md(m), one has the vector bundle

Kc`,d`(1,n`,m`). Let

f` :Mc(n)×Md(m)→Mc`(1,n`)×Md`(1,m`),

denote the canonical projection. Then the vector bundle pullback, f ∗`Kc`,d`(1,n`,m`),

is a vector bundle on the full product Mc(n)×Md(m).

Lemma 4.13. Let X,X1, X2 be smooth algebraic varieties, G,G1, G2 reductive alge-

braic groups acting freely on X,X1, X2, respectively, such that X/G,X1/G1, X2/G2


are smooth, and let V, V1, V2 be representations of G,G1, G2, respectively. Moreover,

suppose that there exists an embedding G1×G2 → G such that V ∼= V1⊕V2 as G1×G2-

representations and that there exists a G1×G2-equivariant morphism ϕ : X1×X2 → X

such that the induced morphism ϕ : X1/G1 ×X2/G2 → X/G is an isomorphism. Let

fi : X1/G1 ×X2/G2 → Xi/Gi denote the projection. Then the vector bundles

f ∗1 (V1 ×G1 X1)⊕ f ∗2 (V2 ×G2 X2)→ X1/G1 ×X2/G2, and V ×G X → X/G,

are isomorphic.

Proof: For simplicity, we will view G1 × G2 as a subgroup of G and identify V

with V1 ⊕ V2, as G1 ×G2-modules. Define

ψ : (V1 ×X1)⊕ (V2 ×X2)→ V ×X,

(v1, x1)⊕ (v2, x2) 7→ (v1 ⊕ v2, ϕ(x1, x2)).

It is clear that ψ is a morphism of varieties. Moreover, one checks that the following

diagram commutes:

(G1 ×G2)× ((V1 ×X1)⊕ (V2 ×X2)) (V1 ×X1)⊕ (V2 ×X2)

G× (V ×X) V ×X

i× ψ ψ,

where i : G1 ×G2 → G is the inclusion map and the horizontal arrows represent the

group action. So, by Lemma 2.3, we have an induced morphism ψ : (V1 ×G1 X1) ⊕

(V2 ×G2 X2)→ V ×G X, which yields the following commutative diagram:

(V1 ×G1 X1)⊕ (V2 ×G2 X2) V ×G X

X1/G1 ×X2/G2 X/G.

ψ

ϕ


Since, by assumption, ϕ is an isomorphism, it remains only to show that ψ is an

isomorphism and that it induces linear maps on the corresponding fibers. The latter

is obvious, and so we focus on the former. By virtue of X/G, X1/G1 and X2/G2 being

smooth, we have that (V1 ×G1 X1)⊕ (V2 ×G2 X2) and V ×G X are smooth. Thus, by

Lemma 2.11, it suffices to show that ψ is bijective and etale. To show bijectivity, it

suffices to look at the fibers. For every (G1 · x1, G2 · x2) ∈ X1/G1×X2/G2, the linear

map induced by ψ on the fibers is given by

(V1 ×G1 G1 · x1)⊕ (V2 ×G2 G2 · x2)→ V ×G G · ϕ(x1, x2),

G1 · (v1, x1)⊕G2 · (v2, x2) 7→ G · (v1 ⊕ v2, ϕ(x1, x2)),

which is clearly bijective. To show that ψ is etale, by Corollary 2.6,

TG1·(v1,x1)⊕G2·(v2,x2)

(⊕k=1,2

(Vk ×Gk Xk)

)∼=⊕k=1,2

T(vk,xk)(Vk ×Xk)/ im dσ(vk,xk)

∼=⊕k=1,2

(Tvk(Vk)⊕ Txk(Xk)) / im dσ(vk,xk),

and

TG·(v1⊕v2,ϕ(x1,x2))(V ×G X) ∼= T(v1⊕v2,ϕ(x1,x2))(V ×X)/ im dσ(v1⊕v2,ϕ(x1,x2))

∼=(T(v1⊕v2)(V )⊕ Tϕ(x1,x2)(X)

)/ im dσ(v1⊕v2,ϕ(x1,x2)).

The differential dψ is given by

((y1 ⊕ z1) + im dσ(v1,x1))⊕ ((y2 ⊕ z2) + im dσ(v2,x2))

7→ ((y1 ⊕ y2)⊕ dϕ(z1, z2)) + im dσ(v1⊕v2,ϕ(x1,x2)).

We will first show that dψ is surjective. Let (y′⊕z′)+im dσ(v1⊕v2,ϕ(x1,x2)) ∈ (Tv1⊕v2(V )⊕

Tϕ(x1,x2)(X))/ im dσ(v1⊕v2,ϕ(x1,x2)). Since dϕ is an isomorphism, there exists a (z1 +

im dσx1)⊕ (z1 + im dσx2) ∈ Tx1(X1)/ im dσx1 ⊕ Tx2(X2)/ im dσx2 such that

dϕ((z1 + im dσx1)⊕ (z1 + im dσx2)) = dϕ(z1, z2) + im dσϕ(x1,x2) = z′ + im dσϕ(x1,x2).


Equivalently, z′ − dϕ(z1, z2) = dσϕ(x1,x2)(h), for some h ∈ T1(G). We also note that

σ(v1⊕v2,ϕ(x1,x2)) : G→ V ×X is the restriction of

σv1⊕v2 × σϕ(x1,x2) : G×G→ V ×X,

to the diagonal of G×G. Hence,

(y′, z′) + dσ(v1⊕v2,ϕ(x1,x2)) = ((y′ − dσv1⊕v2(h))⊕ ϕ(z1, z2)) + dσ(v1⊕v2,ϕ(x1,x2)).

Thus, (y′, z′) + dσ(v1⊕v2,ϕ(x1,x2)) ∈ im dψ. By dimension counting, dψ is an isomor-

phism, and so ψ is etale.

Lemma 4.14. There is an isomorphism of vector bundles

Kc,d(n,m) ∼=s⊕`=1

f ∗` (Kc`,d`(1,n`,m`)).

Proof: We identify each point

([A11, B

11 , i

11], . . . , [As1, B

s1, i

s1], [A1

2, B12 , i

12], . . . , [As2, B

s2, i

s2]) ∈Mc(n)×Md(m),

with its image, ([A1, B1, i1, 0], [A2, B2, i2, 0]) ∈Mc(s, |n|)T•×Md(s, |m|)T• , under the

isomorphism given in Equation (4.9). By Lemma 4.13,

Vk|Mc(n)×Md(m)∼=

s⊕`=1

f ∗` V`k,

where V`k = Vk(c`,d`, 1,n`,m`). Likewise,

Wk|Mc(n)×Md(m)∼=

s⊕`=1

f ∗`W`k,

where W`k = Wk(c`,d`, 1,n`,m`). Thus, Kc,d(s, |n|, |m|) restricted to Mc(n) ×


Md(m) is the middle cohomology of

s⊕k,`=1

Hom(f ∗kVk1 , f ∗` V`2)ζres−−→

s⊕k,`=1

Hom(f ∗kVk1 , f ∗` V`2)⊕ Hom(f ∗kVk1 , f ∗` V`2)

⊕

Hom(f ∗kWk1 , f

∗` V`2)⊕ Hom(f ∗kVk1 , f ∗`W`

2)

τres−−→

s⊕k,`=1

Hom(f ∗kVk1 , f ∗` V`2),

(4.13)

where ζres and τres are the restrictions of ζ and τ to the indicated domains (note

that since we only consider the action of T•, the “t” and “t−1” terms from Complex

(4.12) may be safely omitted). It is easy to see that we have the following equality of

sections:

Ak =s⊕`=1

f ∗`A`k, Bk =

s⊕`=1

f ∗`B`k, ik =

s⊕`=1

f ∗` i`k,

for k = 1, 2. Therefore, Complex (4.13) decomposes as a direct sum of complexes:

s⊕k,`=1

Hom(f ∗kVk1 , f ∗` V`2)ζk`−→

Hom(f ∗kVk1 , f ∗` V`2)⊕ Hom(f ∗kVk1 , f ∗` V`2)

⊕

Hom(f ∗kWk1 , f

∗` V`2)⊕ Hom(f ∗kVk1 , f ∗`W`

2)

τk`−→ Hom(f ∗kVk1 , f ∗` V`2)),

where

ζk`(X) =

Xf ∗k (Ak1)− f ∗` (A`2)X

Xf ∗k (Bk1 )− f ∗` (B2)X

Xf ∗k (ik1)

0

, and

τk`

C

D

a

b

= f ∗` (A`2)D −Df ∗k (Ak1) + Cf ∗k (Bk1 )− f ∗` (B`

2)C + f ∗` (i`2)b.


We therefore have that

Kc,d(n,m) =s⊕

k,`=1

(ker τk`/ im ζk`)T• .

The bundle ker τk`/ im ζk` is the vector bundle pullback of Kck,d`(1,nk,m`) by the

projection

Mc(n)×Md(m)→Mck(1,nk)×Md`(1,m`).

Thus the task of computing (ker τk`/ im ζk`)T• , reduces to computingKck,d`(1,nk,m`)

T• .

The T•-fixed points of Kck,d`(1,nk,m`) may be computed (as a set) by comput-

ing the T•-fixed points of the fibers. Over the point ([Ak1, Bk1 , i

k1], [A`2, B

`2, i

`2]) ∈

Mck(1,nk)×Md`(1,m`), we may identify the fiber of Vk1 with Cnk by

(Cnk ×GLnk(C) (Ak1, B

k1 , i

k1))× [A`2, B

`2, i

`2]→ Cnk ,(

GLnk(C) · (v, (Ak1, Bk1 , i

k1)), [A`2, B

`2, i

`2])7→ v.

Similarly, we identify the fiber of V`2 with Cm` , the fiber of Wk1 with C1k and the

fiber of W`2 with C1`. Via these identifications, the fiber of Kc,d(n,m) is the middle

cohomology of

Hom(Cnk ,Cm`)ζk`−→

Hom(Cnk ,Cm`)⊕ Hom(Cnk ,Cm`)

⊕

Hom(C1k,Cm`)⊕ Hom(Cnk ,C1`)

τk`−→ Hom(Cnk ,Cm`),

where here we view ζk` and τk` as linear maps. By Lemma 2.10, (ker τk`/ im ζk`)T• ∼=

ker τk`|T•/ im ζk`|T• , where ζk`|T• and τk`|T• are the restrictions of ζk` and τk` to the

T•-fixed points of their respective domains. Via our identification of Vk1 with Cnk , we

have that T• acts on Cnk via multiplication by zk−1 for all z ∈ T•. That is,

z · v = zk−1v,

for all z ∈ T• and v ∈ Cnk . Likewise, T• acts on Cm` by multiplication by z`−1, on

C1k by multiplication by zk−1, and on C1` by multiplication by z`−1, for all z ∈ T•.


Therefore, for all z ∈ T• and f ∈ Hom(Cnk ,Cm`),

z · f = zk−`f.

Thus,

Hom(Cnk ,Cm`)T• =

Hom(Cnk ,Cmk), if k = `,

0, if k 6= `.

Similarly, Hom(C1k,Cm`) and Hom(Cnk ,C1`) are zero when k 6= `. Since the fibers

of Kck,d`(1,nk,m`)T• are all zero when k 6= `, we conclude that Kck,d`(1,nk,m`)

T• = 0

for all k 6= `. Therefore,

Kc,d(n,m) =s⊕`=1

f ∗`Kc`,d`(1,n`,m`).

From our diagram of inclusions, (4.10), we have that

Mc(n)×Md(m) ⊇Mc(n)ZR ×Md(m)ZR

∼=∐|v`|=n`|u`|=m`

Mc(v1, . . . ,v`)×Md(u1, . . . ,us).

We may thus view Mc(v1, . . . ,vs) × Md(u1, . . . ,us) as a subvariety of Mc(n) ×

Md(m), and consider the restriction of Kc,d(n,m):

Kc,d(v1,u1, . . . ,vs,us) := Kc,d(n,m)|Mc(v1,...,vs)×Md(u1,...,us),

which is a vector bundle on Mc(v1, . . . ,vs)×Md(u1, . . . ,us). In the following section,

the vector bundles Kc,d(n,m) and Kc,d(v1,u1, . . . ,vs,us) will allows us to construct

geometric versions of the Heisenberg algebra, the Clifford algebra and glr.

Chapter 5

Geometric Realizations of the

Basic Representation

In this chapter, we present the main theorem of this paper, Theorem 5.21, which

describes our geometric realizations of the basic representation of glr. We will do so

by constructing the oscillator algebra, the Clifford algebra and glr as geometric oper-

ators (in the sense of (4.4)). Using the Localization Theorem, we will consider these

geometric operators as operators on the same cohomology. We will show that these

operators satisfy the same relations as those observed by their algebraic counterparts

in Lemma 1.21, and that the cohomology on which they act (or more specifically the

full C-lattice A) corresponds naturally to fermionic Fock space.

Throughout this chapter, for any T?-equivariant vector bundle E, we let ctnv(E)

denote the top nonvanishing T?-equivariant Chern class of E. We will also frequently

make use of the Kunneth formula

H∗T?(X × Y ) ∼= H∗T?(X)⊗H∗T?(Y ),

and simply identify the two rings.

We begin by constructing the geometric version of the oscillator/Heisenberg alge-

bra. The dimension ofMc(n) is 4|n|, thus elements of H2(|n|+|m|)T?

(Mc(n)×Md(m))

83

5. Geometric Realizations of the Basic Representation 84

will define operators H2|n|T?

(Mc(n)) → H2|m|T?

(Md(m)). Consider the vector bundle

Kc,d(n,m) overMc(n)×Md(m). The rank of Kc,d(n,m) is |n|+ |m|, which can be

seen from the following lemma.

Lemma 5.1. Let (I,J) ∈ Mc(n)T? ×Md(m)T?. The equivariant Euler class of the

restriction of Kc,d(n,m) to (I,J) is

eT?(Kc,d(n,m)(I,J)) =s∏`=1

∏x∈λ(I`)

(d` − c` − hλ(I`),λ(J`)(x))R

r`ε

×

∏y∈λ(J`)

(d` − c` + hλ(J`),λ(I`)(y))R

r`ε

.

Proof: By Lemma 4.14,

Kc,d(n,m) ∼=s⊕`=1

f ∗` (Kc`,d`(1,n`,m`)),

and so, by properties of the Euler class,

eT?(Kc,d(n,m)) =s∏`=1

f ∗` (eT?(Kc`,d`(1,n`,m`)))

=s∏`=1

(1⊗`−1 ⊗ eT?(Kc`,d`(1,n`,m`))⊗ 1⊗s−`

).

Now, by setting r = 1 and replacing ε by (R/r`)ε in [15, Lemma 3.3], we have that

eT?(Kc`,d`(1,n`,m`)(I`,J`)) =

∏x∈λ(I`)

(d` − c` − hλ(I`),λ(J`)(x))R

r`ε

×

∏y∈λ(J`)

(d` − c` + hλ(J`),λ(I`)(y))R

r`ε

.

The result follows.


Recall from [15, Section 3.3], that the top nonvanishing Chern class ofKc,c(1, n,m)

is

ctnv(Kc,c(1, n,m)) =

c2n(Kc,c(1, n, n)), if n = m,

cn+m−1(Kc,c(1, n,m)) if n 6= m.

(5.1)

By Lemma 4.14, the top nonvanishing Chern class of Kc,c(n,m) may be computed

by

ctnv(Kc,c(n,m)) = ctnv

(s⊕`=1

f ∗` (Kc`,c`(1,n`,m`))

)=

s∏`=1

f ∗` (ctnv(Kc`,c`(1,n`,m`)))

=s∏`=1

(1⊗`−1 ⊗ ctnv(Kc`,c`(1,n`,m`))⊗ 1⊗s−`

).

Therefore, if n = m,

ctnv(Kc,c(n,n)) = c2|n|(Kc,c(n,n)),

whereas if n differs from m in exactly one component,

ctnv(Kc,c(n,m)) = c|n|+|m|−1(Kc,c(n,m)).

For each ` = 1, . . . , s, define α` ∈ H∗T?(Mc`(1,n`)T?×Md`(1,m`)

T?) =⊕

(I,J)H∗T?(pt)

to be the element with (I, J)-th component

α`(I,J) =

ε

eT? (T(I,J)), if n` = m`,

0, otherwise,

where T(I,J) is the tangent space of (I, J) inMc`(1,n`)×Md`(1,m`). Let α` := i∗(α`),

where i : Mc`(1,n`)T? × Md`(1,m`)

T? → Mc(1,n`) × Md(1,m`) is the natural

inclusion. Denote by iI,J the inclusion (I, J) →Mc(1,n`)×Md(1,m`). Then by

[5, Equation (9.3)],

i∗I,J(α`) = (i∗I,J i∗)(α`) = eT?(T(I,J)) ∪ α`(I,J) =

ε, if n` = m`,

0, otherwise.

Let γ` := f ∗` (α`) = 1⊗`−1 ⊗ α` ⊗ 1⊗s−` ∈ H∗T?(Mc(n)×Md(m)). Note that γ` is an

element of degree 2.


Definition 5.2 (Geometric oscillator/Heisenberg operators). For ` = 1, . . . , s and

k ∈ Z, define operators

P`(k) :⊕n,c

H2|n|T?

(Mc(n))→⊕n,c

H2|n|T?

(Mc(n)),

by

P`(k)|H2|n|T?

(Mc(n))=

(R/r`)γ` ∪ ctnv(Kc,c(n,n− k1`)), if k < 0,

−(R/r`)γ` ∪ ctnv(Kc,c(n,n− k1`)), if k > 0,

∈ H2(2|n|−k)T (Mc(n)×Mc(n− k1`))

P`(0)|H2|n|T?

(Mc(n))= c` · ctnv(Kc,c(n,n)) = c` id .

The P`(k) will be called geometric oscillators (or, for k 6= 0, geometric Heisenberg

operators).

Theorem 5.3. The operators P`(k) preserve the space A and satisfy the commutation

relations

[P`(k),Pm(0)] = 0, and [P`(k),Pm(j)] =1

kδ`,mδk+j,0 id, k 6= 0.

In particular, the mapping

P`(k) 7→ P`(k),

defines a representation of the s-coloured oscillator algebra on A and the linear map

determined by

[I] 7→(sλ(I1) ⊗ qc(I1)

)⊗ · · · ⊗

(sλ(Is) ⊗ qc(Is)

),

is an isomorphism of s-coloured oscillator algebra representations A → B. This

isomorphism maps A(c)→ B(c), for all c ∈ Z.

Proof: Take m = n− k1`. We know that

ctnv(Kc,c(n,m)) =∏s

i=1 (1⊗i−1 ⊗ ctnv(Kci,ci(1,ni,mi))⊗ 1⊗s−i) .


By [15, Lemma 3.10], for i 6= `, we have that ctnv(Kci,ci(1,ni,ni)) is the identity as an

operator H2niT?

(Mci(1,ni))→ H2niT?

(Mci(1,ni)), i.e. ctnv(Kci,ci(1,ni,mi)) = 1. Thus,

ctnv(Kc,c(n,m)) = 1⊗`−1 ⊗ ctnv(Kc`,c`(1,n`,n` − k))⊗ 1⊗s−`.

Therefore, via the Kunneth formula,

⊕n,cH

2|n|T?

(Mc(n)) ∼=⊕

n,cH2n1T?

(Mc1(1,n1))⊗ · · · ⊗ H2nsT?

(Mcs(1,ns)),

we have

P`(k)|H2|n|T?

(Mc(n))= 1⊗(`−1) ⊗ (R/r`)α

` ∪ ctnv(Kc`,c`(1,n`,n` − k))⊗ 1⊗(s−`).

By [15, Theorem 3.14], for a fixed ` ∈ 1, . . . , s, the operators P`(k) preserve the

space⊕

n`,c`Ac`(n`) (see Remark 4.8) and satisfy the 1-coloured oscillator algebra

relations. The general result then follows by extension to the whole tensor product.

For the geometric version of the Clifford algebra, we recall from [15, Section 3.2]

that

ctnv(Kc,c±1(1, n,m)) = cn+m(Kc,c±1(1, n,m)). (5.2)

Therefore, by equations (5.1) and (5.2), if ni = mi for all i 6= `, then

ctnv(Kc,c±1`(n,m)) = c|n|+|m|(Kc,c±1`(n,m)).

Definition 5.4 (Geometric Clifford operators). For ` = 1, . . . , s and k ∈ Z, define

operators

Ψ`(k),Ψ∗`(k) :⊕n,c

H2|n|T?

(Mc(n))→⊕n,c

H2|n|T?

(Mc(n)),

by

Ψ`(k)|H2|n|T?

(Mc(n)):= (−1)c1+···+c`−1ctnv(Kc,c+1`(n,n + (k − c` − 1)1`))


∈ H2(2|n|+k−c`−1)T?

(Mc(n)×Mc+1`(n + (k − c` − 1)1`)),

Ψ∗`(k)|H2|n|T?

(Mc(n)):= (−1)c1+···+c`−1ctnv(Kc,c−1`(n,n− (k − c`)1`))

∈ H2(2|n|−k+c`)T?

(Mc(n)×Mc−1`(n− (k − c`)1`)).

The Ψ`(k) and Ψ∗`(k) will be called geometric Clifford operators.

Theorem 5.5. The operators Ψ`(k) and Ψ∗`(k) preserve the space A and satisfy the

relations

Ψ`(k),Ψ∗j(i) = δkiδ`j, Ψ`(k),Ψj(i) = Ψ∗`(k),Ψ∗j(i) = 0.

In particular, the mapping

ψ`(k) 7→ Ψ`(k), ψ∗` (k) 7→ Ψ∗`(k),

defines a representation of the s-coloured Clifford algebra on A and the linear map

determined by

[I] 7→ I,

is an isomorphism of Clifford algebra representations A → F. This isomorphism

maps A(c)→ F(c), for all c ∈ Z.

Proof: Using an argument analogous to the proof of Theorem 5.3, one can show

that

Ψ`(k)|HT? (Mc(n)) = (−1)c1+···+c`−1(1⊗`−1⊗ctnv(Kc`,c`+1(1,n`,n`+k−c`−1))⊗1⊗s−`),

Ψ∗`(k)|HT? (Mc(n)) = (−1)c1+···+c`−1(1⊗`−1 ⊗ ctnv(Kc`,c`−1(1,n`,n` − k + c`))⊗ 1⊗s−`).

By [15, Theorem 3.6], for fixed `, the operators (−1)c1+···+c`−1ctnv(Kc`,c`+1(1,n`,n` +

k − c` − 1)) and (−1)c1+···+c`−1ctnv(Kc`,c`−1(1,n`,n` − k + c`)) preserve the space⊕n`,c`

Ac`(n`) (see Remark 4.8) and satisfy the 1-coloured Clifford algebra relations.


The general result then follows by extension to the whole tensor product.

Recall from Diagram 4.10 that, since

Mc(n)ZR ∼=∐|v`|=n`

Mc(v1, . . . ,vs),

we may view Mc(v1, . . . ,vs)×Md(u1, . . . ,us) as a subvariety of Mc(n)×Md(m),

where n` = |v`| and m` = |u`|. We therefore have an action of T? onMc(v1, . . . ,vs)

and ∐v`

Mc(v1, . . . ,vs)T? =

∐n

Mc(n)T? .

Let

Kc(n; `, k)± :=∐|vi|=ni

Kc,c(v1,v1, . . . ,v`,v` ± 1k, . . . ,v

s,vs),

which is a vector bundle on∐|vi|=ni

Mc(v1, . . . ,vs)×Mc(v

1, . . . ,v`±1k, . . . ,vs). By

construction, for any (I,J) ∈Mc(v1, . . . ,vs)T? ×Mc(v

1, . . . ,v` ± 1k, . . . ,v`)T? ,

ctnv(Kc(n; `, k)±(I,J)) = ctnv(Kc,c(n,n± 1`)(I,J)).

Thus,

ctnv(Kc(n; `, k)±) = c(2|n|±1)−1(Kc(n; `, k)±).

By the same reasoning, for (I,J) ∈Mc(v1, . . . ,vs)T? ×Mc(v

1, . . . ,vs)T? ,

ctnv(Kc,c(v1,v1, . . . ,vs,vs)(I,J)) = ctnv(Kc,c(n,n)(I,J)),

and so

ctnv(Kc,c(v1,v1, . . . ,vs,vs)) = c2|n|(Kc,c(v

1,v1, . . . ,vs,vs)).

Let h :∐

v`,u` Mc(v1, . . . ,vs) × Md(u1, . . . ,us) ∼= Mc(n)ZR × Md(m)ZR →

Mc(n)×Md(m) denote the natural inclusion and let β` := h∗(γ`) for all ` = 1, . . . , s.

For ` = 1, . . . , s and k = 0, . . . , r` − 1, we define

E`k(c,n) := −(R/r`)β` ∪ ctnv(Kc(n; `, k)−)


∈ H4|n|−2T?

∐|vi|=ni

Mc(v1, . . . ,vs)×Mc(v

1, . . . ,v` − 1k, . . . ,vs)

,

F`k(c,n) := (R/r`)β` ∪ ctnv(Kc(n; `, k)+)

∈ H4|n|+2T?

∐|vi|=ni

Mc(v1, . . . ,vs)×Mc(v

1, . . . ,v` + 1k, . . . ,vs)

,

H`k(c; v1, . . . ,vs) :=

((1c)k −

r`−1∑j=0

a`kjv`j

)ctnv(Kc,c(v

1,v1, . . . ,vs,vs))

∈ H4(|v1|+···+|vs|)T?

(Mc(v1, . . . ,vs)×Mc(v

1, . . . ,vs)),

where a`kj is the (k, j)-th entry of the generalized Cartan matrix of type Ar`−1, i.e.

a`kj =

2, if k = j,

−1, if k = j ± 1,

0, otherwise,

if r` ≥ 2,

a`kj =

2, if k = j,

−2 if k = j ± 1,

if r` = 2,

a`00 = 0, if r` = 1.

Note that the elements E`k, F`k and H`k do not define operators since they do not

lie in the middle degree cohomology of their respective quiver varieties. However,

we can push these elements forward to the middle degree cohomology of the appro-

priate moduli spaces; we describe this process below. For what follows, it will be

convenient to view elements of H∗T?(Mc(v1, . . . ,vs)×Md(u1, . . . ,us)) as elements of

H∗T?(Mc(n)ZR ×Md(m)ZR). That is, by identifying

H∗T?(Mc(n)ZR ×Md(m)ZR) ∼= H∗T?

∐|vi|=ni|ui|=mi

Mc(v1, . . . ,vs)×Md(u1, . . . ,us)


∼=⊕|vi|=ni|ui|=mi

H∗T?(Mc(v1, . . . ,vs)×Md(u1, . . . ,us)),

we will identify elements of H∗T?(Mc(v1, . . . ,vs)×Md(u1, . . . ,us)) with their images

under the inclusion

H∗T?(Mc(v

1, . . . ,vs)×Md(u1, . . . ,us))→ H∗T?(Mc(n)ZR ×Md(m)ZR).

In this way, we will consider

E`k(c,n) ∈ H∗T?(Mc(n)ZR ×Mc(n− 1`)ZR),

F`k(c,n) ∈ H∗T?(Mc(n)ZR ×Mc(n + 1`)ZR),

H`k(c; v1, . . . ,vs) ∈ H∗T?(Mc(n)ZR ×Mc(n)ZR).

Recall from the Localization Theorem (4.1) that we have isomorphisms f1 and f2 as

in the diagram below:

H∗T?(Mc(n)×Md(m))

H∗T?(Mc(n)T? ×Md(m)T?).

H∗T?(Mc(n)ZR ×Md(m)ZR)

f1 f2

Let

µ : H∗T?(Mc(n)T? ×Md(m)T?)→ H∗T?(Mc(n)T? ×Md(m)T?),

be the C(b1, . . . , bs, ε)-linear map determined by

1(I,J) 7→eT?(T ′I,J)

eT?(TI,J)1(I,J),

where TI,J and T ′I,J are the tangent spaces of (I,J) inMc(n)×Md(m) andMc(n)ZR×

Md(m)ZR , respectively. Let η : H∗T?(Mc(n)ZR×Mc(m)ZR)→ H∗T?(Mc(n)×Md(m))

be the composition

η := f−11 µ f2. (5.3)


Lemma 5.6. The map η is degree-preserving. In particular, the images of E`k(c,n),

F`k(c,n) and H`k(c; v1, . . . ,vs) under η lie in the middle degree cohomology ofMc(n)×

Mc(n∓ 1`) and Mc(n)×Mc(n), respectively.

Proof: The map f2 decreases the degree by rk(eT?(T ′I,J)). The map µ increases

the degree by rk(eT?(T ′I,J))− rk(eT?(TI,J)). Finally, the map f−11 increases the degree

by rk(eT?(TI,J)). Thus, the composition of these maps is degree preserving.

The second statement in the lemma follows from the fact that E`k(c,n), F`k(c,n)

and H`k are elements of degree 4|n| − 2, 4|n|+ 2 and 4|n|, respectively.

Definition 5.7 (Geometric diagonal Chevalley operators). For ` = 1, . . . , s and k =

0, . . . , r` − 1, define operators

E`k,F

`k,H

`k :⊕n,c

H2|n|T?

(Mc(n))→⊕n,c

H2|n|T?

(Mc(n)),

by

E`k|H2|n|

T?(Mc(n))

:= η(E`k(c,n)) ∈ H4|n|−2T?

(Mc(n)×Mc(n− 1`)),

F`k|H2|n|

T?(Mc(n))

:= η(F`k(c,n)) ∈ H4|n|+2T?

(Mc(n)×Mc(n + 1`)),

H`k|H2|n|

T?(Mc(n))

:=∑|vi|=ni

η(H`k(c; v1, . . . ,vs)) ∈ H4|n|

T?(Mc(n)×Mc(n)).

The E`k, F`

k and H`k will be called geometric (diagonal) Chevalley operators.

Our next goal will be to describe the geometric Chevalley operators in terms of

geometric Clifford operators. To that end, we prove the following useful lemma.

Lemma 5.8. Let (I,J) ∈ Mc(n)T? × Md(m)T? and β ∈ H2kT?

(Mc(v1, . . . ,vs) ×

Md(u1, . . . ,us)). Let

Mc(n)×Md(m)i1×i2←−−−Mc(n)T? ×Md(m)T?

j1×j2−−−→Mc(n)ZR ×Md(m)ZR ,


be the natural inclusions and let (i1 × i2)(I,J) and (j1 × j2)(I,J) denote the restrictions

of i1 × i2 and j1 × j2, respectively, to (I,J). If

(I,J) ∈Mc(v1, . . . ,vs)T? ×Md(u1, . . . ,us)T? ,

then

〈η(β ∪ c|n|+|m|−k(Kc,d(v1,u1, . . . ,vs,us))[I], [J]〉 = 〈β′ ∪ c|n|+|m|−k(Kc,d(n,m))[I], [J]〉,

where β′ is a preimage of (j1 × j2)∗(I,J)(β) under the map (i1 × i2)∗(I,J). Otherwise,

〈η(β ∪ c|n|+|m|−k(Kc,d(v1,u1, . . . ,vs,us)))[I], [J]〉 = 0.

Proof: To simplify notation, we write K = Kc,d(v1,u1, . . . ,vs,us), K = Kc,d(n,m)

and c = c|n|+|m|−k. We begin by explicitly computing η(β∪c(K)) (recall η = f−11 µf2

as in (5.3)). By definition of f2,

f2(β ∪ c(K)) =

((j1 × j2)∗(K,L)(β ∪ c(K))

eT?(T ′(K,L))

)(K,L)∈Mc(n)T?×Md(m)T?

=

(βK,L ∪ (j1 × j2)∗(K,L)(c(K))

eT?(T ′(K,L))


,

where T ′(K,L) is the tangent space of (K,L) in Mc(n)ZR ×Mc(m)ZR , and βK,L =

(j1 × j2)∗(K,L)(β). By applying µ, we get

µ f2(β ∪ c(K)) =

(βK,L ∪ (j1 × j2)∗(K,L)(c(K))

eT?(T(K,L))


,

where T(K,L) is the tangent space of (K,L) in Mc(n) ×Md(m). By definition, the

map f−11 = (i1 × i2)∗, and thus,

η(β ∪ c(K)) = (i1 × i2)∗

(βK,L ∪ (j1 × j2)∗(K,L)(c(K))

eT?(T(K,L))


.


By definition of the bilinear form,

〈η(β ∪ c(K))[I], [J]〉 = (−1)|m|p∗((i1 × i2)∗)−1(η(β ∪ c(K)) ∪ [I]⊗ [J])

= (−1)|m|p∗((i1 × i2)∗)−1

(η(β ∪ c(K))

eT?(T −I )eT?(T −J )∪ (i1 × i2)∗(1(I,J))

),

where the last equality comes from the definition of [I] and [J]. By the projection

formula,

〈η(β ∪ c(K))[I], [J]〉 = (−1)|m|p∗

((i1 × i2)∗(η(β ∪ c(K)))

eT?(T −I )eT?(T −J )∪ 1(I,J)

).

By [5, Equation 9.3], (i1 × i2)∗(i1 × i2)∗ is simply multiplication by the Euler class of

the tangent space. Hence,

(i1 × i2)∗η(β ∪ c(K)) =(βK,L ∪ (j1 × j2)∗(K,L)(c(K))


.

Thus,

〈η(β ∪ c(K))[I], [J]〉 = (−1)|m|p∗

(βI,J ∪ (j1 × j2)∗(I,J)(c(K))

eT?(T −I )eT?(T −J )

).

By construction, if

(I,J) /∈Mc(v1, . . . ,vs)T? ×Md(u1, . . . ,us)T? ,

then

(j1 × j2)∗(I,J)(c(K)) = 0.

On the other hand, if

(I,J) ∈Mc(v1, . . . ,vs)T? ×Md(u1, . . . ,us)T? ,

then by functoriality of the Chern class and the construction of K,

(j1 × j2)∗(I,J)(c(K)) = c(K(I,J)) = c(K(I,J)).

Therefore,

〈η(β∪c(K))[I], [J]〉 = (−1)|m|βI,J ∪ c(K(I,J))

eT?(T −I )eT?(T −J )=

βI,J ∪ c(K(I,J))

eT?(T −I )eT?(T +J )

= 〈β′∪c(K)[I], [J]〉,


where the last equality follows from Lemma 4.9.

Corollary 5.9. Let (I,J) ∈Mc(n)T? ×Mc(m)T?.

1. For m = n− 1`, if

(I,J) ∈Mc(v1, . . . ,vs)T? ×Mc(v

1, . . . ,v` − 1k, . . . ,vs)T? ,

then

〈E`k[I], [J]〉 = 〈P`(1)[I], [J]〉.

Otherwise

〈E`k[I], [J]〉 = 0.

2. For m = n + 1`, if

(I,J) ∈Mc(v1, . . . ,vs)T? ×Mc(v

1, . . . ,v` + 1k, . . . ,vs)T? ,

then

〈F`k[I], [J]〉 = 〈P`(−1)[I], [J]〉.

Otherwise

〈F`k[I], [J]〉 = 0.

3. For m = n,

〈H`k[I], [J]〉 =

((1c`)k −

∑j

a`kjv`j

)δI,J.

Proof: Statements (1) and (2) follow directly from Lemma 5.8 and the definitions

of P`(±1) (note that, in the notation of Lemma 5.8, (i1 × i2)∗(γ`) = (j1 × j2)∗(β`)).

The third statement follows from Lemma 5.8 and the fact that ctnv(Kc,c(n,n)) = id

as an operator on H2|n|T?

(Mc(n)) (see Definition 5.2).


Proposition 5.10. Let (I,J) ∈ Mc(n)T? ×Mc(n + 1`)T?. If λ(Iα) = λ(Jα) for all

α 6= ` and λ(J`) can be obtained by adding one box to λ(I`), then

〈P`(−1)[I], [J]〉 = 1.

Otherwise, 〈P`(−1)[I], [J]〉 = 0.

Proof: This is a direct result of [15, Proposition 5.3 and the proof of Theorem

3.14]. Note that in our case, λ(J`)− λ(I`) consists of a single box.

Lemma 5.11. If I ∈ Mc(v1, . . . ,vs)T?, then v`k is equal to the number of boxes in

λ(I`) whose residue is congruent to k − c` modulo r`.

Proof: Let I` = [A,B, i] ∈ Mc`(1,n`)T? . By [24, Proposition 2.9], λ(I`) is

obtained from I` by drawing a box in the (p, q)-th position if Ap−1Bq−1i 6= 0 (note

that our Young diagrams are rotated 90 clockwise from those in [24]). From Equation

(3.8),

v`k = dim spanApBqi | q − p ≡ k − c` mod r`.

Since the nonzero ApBqi are linearly independent, the boxes (p, q) ∈ λ(I`) whose

residue is congruent to k−c` mod r` are in one-to-one correspondence with a basis of

spanApBqi | q−p ≡ k−c` mod r`. Thus, v`k is equal to the number of such boxes.

Lemma 5.12. Let (I,J) ∈Mc(n)T? ×Mc(n + 1`)T?. Then

〈F`k[I], [J]〉 = 1,

if λ(Iα) = λ(Jα) for all α 6= ` and λ(J`) can be obtained from λ(I`) by adding one box

whose residue is congruent to k − c` modulo r`. Otherwise, 〈F`k[I], [J]〉 = 0.


Proof: This follows immediately from Corollary 5.9, Proposition 5.10 and Lemma

5.11.

Proposition 5.13. Let (I,J) ∈Mc(n)T? ×Mc(n + 1`)T?. Then for all i ∈ Z,

〈Ψ`(i)Ψ∗`(i− 1)[I], [J]〉 =

1, if J = ψ`(i)ψ∗` (i− 1)I,

0, otherwise.

Proof: This follows immediately from Theorem 5.5.

Theorem 5.14. For ` = 1, . . . , s and k = 0, 1, . . . , r` − 1,

E`k =

∑i∈Z

Ψ`(k + ir`)Ψ∗`(k + ir` + 1),

F`k =

∑i∈Z

Ψ`(k + ir` + 1)Ψ∗`(k + ir`),

as operators on⊕

n,cH2|n|T?

(Mc(n)).

Proof: Let I,J be two s-tuples of semi-infinite monomials of charge c. We first

prove that

〈F`k[I], [J]〉 =

⟨∑i∈Z

Ψ`(k + ir` + 1)Ψ∗`(k + ir`)[I], [J]

⟩. (5.4)

If there exists an α 6= ` such that Iα 6= Jα, then both sides of Equation (5.4) are zero

and we are done. Thus, we assume that Iα = Jα for all α 6= `. Write

I` = i1 ∧ i2 ∧ i3 ∧ · · · , and J` = j1 ∧ j2 ∧ j3 ∧ · · · ,

where im, jm ∈ Z. Recall that the number of boxes in the m-th row of λ(I`) is

im − c` + m− 1 (likewise for λ(J`)). Suppose that 〈F`k[I], [J]〉 = 1. Then by Lemma

5.12, λ(J`) is obtained by adding one box of the appropriate residue to λ(I`). This


means that there exists an m ∈ N such that jn = in for all n 6= m and jm = im + 1.

The position of the added box is thus (m, jm− c` +m− 1). The residue of the added

box must be congruent to k − c` modulo r`, and so,

(jm − c` +m− 1)−m = jm − c` − 1 ≡ k − c` mod r`,

or equivalently

jm − k − 1 ≡ 0 mod r`.

Thus, jm = k + ir` + 1, for some i ∈ Z. Now, since jn = in for all n 6= m and

jm = im + 1, we have that

J = ψ`(jm)ψ∗` (jm − 1)I, and J 6= ψ`(j)ψ∗` (j − 1)I, for all j 6= jm.

Hence,⟨∑i∈Z

Ψ`(k + ir` + 1)Ψ∗`(k + ir`)[I], [J]

⟩= 〈Ψ`(jm)Ψ∗`(jm − 1)[I], [J]〉 = 1,

by Proposition 5.13. We have therefore proven that Equation (5.4) holds when

〈F`k[I], [J]〉 = 1.

Suppose now that 〈F`k[I], [J]〉 = 0. Then λ(J`) cannot be obtained by adding one

box to λ(I`). In particular, J 6= ψ`(i)ψ∗` (i− 1)I for all i ∈ Z. Thus,⟨∑

i∈Z

Ψ`(k + ir` + 1)Ψ∗`(k + ir`)[I], [J]

⟩= 0.

Therefore, Equation (5.4) holds in all cases, and so

F`k =

∑i∈Z

Ψ`(k + ir` + 1)Ψ∗`(k + ir`).

Now, since P`(−1) and P`(1) are adoint (see [15, Lemma 3.13]), it follows by

Corollary 5.9 that E`k and F`

k are adoint. Moreover, since Ψ`(i) and Ψ∗`(i) are adjoint

(see [15, Lemma 3.5]), it follows that∑i∈Z

Ψ`(k + ir` + 1)Ψ∗`(k + ir`), and∑i∈Z

Ψ`(k + ir`)Ψ∗`(k + ir` + 1),


are adjoint. Therefore, for all s-tuples of semi-infinite monomials I, J,

〈E`k[I], [J]〉 = 〈[I],F`

k[J]〉 =

⟨[I],∑i∈Z

Ψ`(k + ir` + 1)Ψ∗`(k + ir`)[J]

⟩

=

⟨∑i∈Z

Ψ`(k + ir`)Ψ∗`(k + ir` + 1)[I], [J]

⟩.

Thus,

E`k =

∑i∈Z

Ψ`(k + ir`)Ψ∗`(k + ir` + 1).

Proposition 5.15. For all ` = 1, . . . , s and k = 0, 1, . . . , r` − 1,

[E`k,F

`k] = H`

k.

Proof: If r` = 1, then

E`0 = P`(1), F`

0 = P`(−1), H`0 = id,

and the result follows from Theorem 5.3.

Now, fix an ` ∈ 1, . . . , s such that r` ≥ 2. For any semi-infinite monomial I of

charge c ∈ Z, we will say that a box (p, q) ∈ λ(I) is a k-box if its residue is congruent

to k− cmod r`. We will say that a box (p, q) ∈ λ(I) is k-removable if (p, q) is a k-box

and (p, q) may be removed from λ(I) to create a new Young diagram. Denote the set

of k-removable boxes of λ(I) by R. We will say that a box (p, q) /∈ λ(I) is k-addable

if (p, q) is a k-box and (p, q) may be added to λ(I) to create a new Young diagram.

Denote the set of k-addable boxes of λ(I) by A.

By Theorem 5.14, we have that, for all I ∈Mc(v1, . . . ,vs)T?

E`k[I] =

∑J

[J], and F`k[I] =

∑K

[K],


where the J run over all semi-infinite monomials such that Ji = Ii for all i 6= ` and

J` is obtained from I` by removing a k-removable box, and the K run over all semi-

infinite monomials such that Ki = Ii for all i 6= ` and K` is obtained from I` by

adding a k-addable box. Thus, it is easy to see that

[E`k,F

`k][I] = (|A| − |R|)[I],

where here A and R refer to the sets of k-addable and k-removable boxes of I`,

respectively. Therefore, it suffices to show that

|A|−|R| = (1c)k−r∑j=0

a`kjv`j = δc,k−2v`k+v`k+1+v`k−1 = δc,k+(v`k+1−v`k)+(v`k−1−v`k),

where, of course, the indices of v` are taken modulo r`.

For the remainder of the proof, we will identify I` with its Young diagram λ =

λ(I`). The case where λ is the empty Young diagram is trivial, so we assume λ

consists of at least one box. The k-border of λ is the set

B := k-boxes (p, q) ∈ λ | (p+ 1, q), (p, q + 1), or (p+ 1, q + 1) /∈ λ.

We partition B with respect to the main diagonal of λ as follows:

B = U ∪M ∪ L,

U = (p, q) ∈ B | q > p, M = (p, p) ∈ B, L = (p, q) ∈ B | p > q.

The sets B,U,M and L are illustrated in the following diagram:


L

U

B

M

Here B is the set of k-boxes in the shaded region, U (resp. L) is the set of boxes in B

that lie above (resp. below) the dashed line, and M is the intersection of B with the

dashed line. Let ∆→ (resp. ∆↓) be the subset of B consisting of boxes which have a

right (resp. lower) neighbour. That is,

∆→ = (p, q) ∈ B | (p, q + 1) ∈ λ, and ∆↓ = (p, q) ∈ B | (p+ 1, q) ∈ λ.

Denote by ∆′→ and ∆′↓ the complements of ∆→ and ∆↓, respectively, in B.

We begin with the case where k 6≡ c` mod r` (so that M = ∅). In the portion

of λ above the main diagonal, every (k + 1)-box occurs as the right neighbour of a

k-box. Conversely, every right neighbour of a k-box (if it has one) is a (k + 1)-box.

If a k-box does not have a right neighbour, it must therefore lie in B. Hence, the

number of k-boxes less the number of (k+ 1)-boxes above the main diagonal is equal

to |U ∩ ∆′→|. In the portion of λ below and including the main diagonal, we can

dualize this argument and conclude that the number of (k+ 1)-boxes less the number

of k-boxes is equal to the number of (k + 1)-boxes (on the border) with no lower

neighbour. Any (k + 1)-box not in the first column and without a lower neighbour

has a left neighbour, which must be a k-box and lie in L and ∆→. Hence, the number

of (k + 1)-boxes less the number of k-boxes is equal to |L ∩∆→| + δ, where δ = 1 if


the last box of the first column of λ is a (k + 1)-box and δ = 0 otherwise. Since v`k

(resp. v`k+1) is the number of k-boxes (resp. (k + 1)-boxes) in λ,

v`k+1 − v`k = |L ∩∆→|+ δ − |U ∩∆′→|. (5.5)

By a completely analogous argument,

v`k−1 − v`k = |U ∩∆↓|+ δ′ − |L ∩∆′↓|,

where δ′ = 1 if the last box of the first row of λ is a (k− 1)-box and δ′ = 0 otherwise.

Now, we note that a k-box is k-removable if and only if it has no right or lower

neighbours. Hence, R = ∆′→ ∩∆′↓ and so

|R| = |∆′→ ∩∆′↓| = |B| − |∆→ ∪∆↓|. (5.6)

We can add a k-box at the end of the first row (resp. column) if and only if the last

box of the first row (resp. column) is a (k−1)-box (resp. (k+1)-box). A k-box (p, q),

where p, q 6= 1, is k-addable if and only if both (p−1, q) and (p, q−1) are in λ, which

occurs if and only if (p− 1, q − 1) ∈ ∆→ ∩∆↓. Hence,

|A| = |∆→ ∩∆↓|+ δ + δ′ = |∆→|+ |∆↓| − |∆→ ∪∆↓|+ δ + δ′. (5.7)

Since M = ∅, we have that B = U ∪ L, and so

|∆→/↓| = |U ∩∆→/↓|+ |L ∩∆→/↓| (5.8)

Combining equations (5.6), (5.7) and (5.8), we get

|A| − |R| = |U ∩∆→|+ |L ∩∆→|+ |U ∩∆↓|+ |L ∩∆↓| − |B|+ δ + δ′

= |U | − |U ∩∆′→|+ |L ∩∆→|+ |U ∩∆↓|+ |L| − |L ∩∆′↓| − |B|+ δ + δ′

= −|U ∩∆′→|+ |L ∩∆→|+ |U ∩∆↓| − |L ∩∆′↓|+ δ + δ′

= (v`k+1 − v`k) + (v`k−1 − v`k).


In the case where k ≡ c` mod r`, one only has to make a few modifications to the

above arguments owing to the fact that M now consists of a box. In the portion of

λ above and including the main diagonal, the number of k-boxes less the number of

(k+1)-boxes is |U∩∆′→|+|M∩∆′→|. In the portion of λ below the main diagonal, one

again has that the number of (k+1)-boxes less the number of k-boxes is |L∩∆→|+δ.

Hence,

v`k+1 − v`k = |L ∩∆→|+ δ − |U ∩∆′→| − |M ∩∆′→|.

And again by a completely analogous argument,

v`k−1 − v`k = |U ∩∆↓|+ δ′ − |L ∩∆′↓| − |M ∩∆′↓|. (5.9)

One can compute |A| and |R| exactly as in equations (5.7) and (5.6). The difference

now is that, since M 6= ∅, we have B = U ∪M ∪ L, and so

|∆→/↓| = |U ∩∆→/↓|+ |M ∩∆→/↓|+ |L ∩∆→/↓|.

Therefore, using similar calculations as before,

|A| − |R| = |U | − |U ∩∆′→|+ |L ∩∆→|+ |U ∩∆↓|+ |L| − |L ∩∆′↓| − |B|+ δ + δ′

+ |M ∩∆→|+ |M ∩∆↓|

= |U | − |U ∩∆′→|+ |L ∩∆→|+ |U ∩∆↓||L| − |L ∩∆′↓| − |B|+ δ + δ′

+ |M | − |M ∩∆′→|+ |M | − |M ∩∆′↓|

= −|U ∩∆′→|+ |L ∩∆→|+ |U ∩∆↓| − |L ∩∆′↓|+ δ + δ′ − |M ∩∆′→|

+ |M | − |M ∩∆′↓|

= 1 + (v`k+1 − v`k) + (v`k−1 − v`k),

which completes the proof.

With the operators E`k, F`

k and H`k, we are able to give the following partial

result towards our goal of giving a geometric realization of Vbasic.


Proposition 5.16. The geometric diagonal Chevalley operators preserve the space

A. Moreover, when r` ≥ 2, the operators E`k, F`

k and H`k satisfy the Kac-Moody

relations for slr`. In particular, the mapping

Ek 7→ E`k, Fk 7→ F`

k, Hk 7→ H`k, (5.10)

for k = 0, 1, . . . , r` − 1, defines a representation of slr` on A and the linear isomor-

phism determined by [I] 7→ I, is an isomorphism of slr`-representations A→ F. The

mapping (5.10) together with

I ⊗ tn 7→ |n|r`P`(nr`), and I ⊗ 1 7→ P`(0),

for n ∈ Z − 0, defines a representation of glr` on A and the mapping [I] 7→ I is

then an isomorphism of glr`-representations A→ F.

When r` = 1, the mapping

I ⊗ tn 7→ |n|P`(n), I ⊗ 1 7→ P`(0), c 7→ idA,

for all n ∈ Z− 0, defines a representation of gl1 on A and the mapping [I] 7→ I is

an isomorphism of gl1-representations A→ F.

Proof: The fact that the geometric diagonal Chevalley operators preserve A

is clear from Corollary 5.9. By Theorem 5.5, we have the following commutative

diagram:

A F

A F

Ψ`(i),Ψ∗`(i) ψ`(i), ψ

∗` (i),

(5.11)

for each ` = 1, . . . , s and i ∈ Z. Fix a value of ` ∈ 1, . . . , s such that r` ≥ 2. For

each k = 0, 1, . . . , r` − 1, let

E`k = Er1+···+r`−1+k, F `

k = Fr1+···+r`−1+k, H`k = Hr1+···+r`−1+k.

Then by Lemma 1.21 and Theorem 5.14, we have the following commutative diagram:


A F

A F

E`k,F

`k E`

k, F`k ,

for all k = 0, 1, . . . , r` − 1. By Proposition 5.15, we therefore also have the following

commutative diagram:

A F

A F

H`k H`

k.

Thus, when r` ≥ 2, we have a representation of slr` on A and the map A→ F is an

isomorphism of slr`-representations. The fact that the addition of the mapping

I ⊗ tn 7→ |n|r`P`(nr`), I ⊗ 1 7→ P`(0),

gives us a representation of glr` on A follows from Lemma 1.21 and Theorem 5.3.

The case where r` = 1 is obvious.

Proposition 5.16 shows that the operators E`k, F`

k, H`k and P`(nr`) on A corre-

spond to the operators associated to the (`, `)-th diagonal blocks of glr on F. Thus,

as was the case algebraically, to complete our realization of glr, it remains only to

construct the operators associated to the off-diagonal blocks.

Let c ∈ Zs and i 6= j ∈ 1, . . . , s. Suppose n,m ∈ Ns such that n` = m` for all

` 6= i, j. By equations (5.1) and (5.2), we have that

ctnv(Kc,c+1i−1j(n,m)) = c|n|+|m|(Kc,c+1i−1j(n,m)).

For i = 1, . . . , s− 1 and k ∈ Z, define

K+c (n; i, k) := Kc,c+1i−1i+1

(n,n + ((k + 1)ri − ci − 1)1i − (kri+1 − ci+1 + 1)1i+1),


K−c (n; i, k) := Kc,c+1i+1−1i(n,n + ((k − 1)ri+1 − ci+1)1i+1 − (kri − ci)1i).

We also define

K+c (n; 0, k) := Kc,c+1s−11(n,n + (krs − cs − 1)1s − (kr1 − c1 + 1)11),

K−c (n; 0, k) := Kc,c+11−1s(n,n + (kr1 − c1)11 − (krs − cs)1s).

Definition 5.17 (Geometric off-diagonal Chevalley operators). For i = 0, 1, . . . , s−1,

define operators

E′i,F′i,H

′i :⊕n,c

H2|n|T?

(Mc(n))→⊕n,c

H2|n|T?

(Mc(n)),

by

E′i|H2|n|T?

(Mc(n))= (−1)ci

∑k∈Z

ctnv(K+c (n; i, k)),

F′i|H2|n|T?

(Mc(n))= (−1)ci

∑k∈Z

ctnv(K−c (n; i, k)),

for i = 1, . . . , s− 1, and

E′0|H2|n|T?

(Mc(n))= (−1)c1+···+cs−1

∑k∈Z

ctnv(K+c (n; 0, k)),

F′0|H2|n|T?

(Mc(n))= (−1)c1+···+cs−1

∑k∈Z

ctnv(K−c (n; 0, k)),

and finally, H′i := [E′i,F′i], for all i = 0, . . . , s− 1.

Remark 5.18. For the diagonal Chevalley operators, the space Ac(v1, . . . ,vs) is a

subspace of the the v`-weight space of glr` , thus allowing us to define the operators

H`k explicitly in terms of the v`. For the off-diagonal Chevalley operators, the v` no

longer encode information about the weights of glr. Hence, our defining H′i simply as

the commutator of E′i and F′i seems the best we can achieve.

Lemma 5.19. As operators on A,

E′i =∑k∈Z

Ψi((k + 1)ri)Ψ∗i+1(kri+1 + 1), F′i =

∑k∈Z

Ψi+1((k − 1)ri+1 + 1)Ψ∗i (kri),


for i = 1, . . . , s− 1, and

E′0 =∑k∈Z

Ψs(krs)Ψ∗1(kr1 + 1), F′0 =

∑k∈Z

Ψ1(kr1 + 1)Ψ∗s(krs).

Proof: Fix i ∈ 1, . . . , s− 1. Consider the restriction of E′i to H2|n|T?

(Mc(n)). To

simplify notation, let d = c + 1i−1i+1 and mk = n + ((k+ 1)ri− ci− 1)1i− (kri+1−

ci+1 + 1)1i+1. Then

E′i = (−1)ci∑k∈Z

ctnv(Kc,d(n,mk)) = (−1)ci∑k∈Z

ctnv

(s⊕`=1

f ∗`Kc`,d`(1,n`, (mk)`)

)

= (−1)ci∑k∈Z

(s∏`=1

(1⊗`−1 ⊗ ctnv(Kc`,d`(1,n`, (mk)`))⊗ 1⊗s−`)

),

where the second equality follows from Lemma 4.14 and the third equality is the

application of the Kunneth formula. For ` 6= i, i+ 1,

ctnv(Kc`,d`(1,n`, (mk)`)) = ctnv(Kc`,c`(1,n`,n`)) = 1,

by [15, Lemma 3.10]. Thus,

E′i = (−1)ci∑k∈Z

(1⊗i−1 ⊗ ctnv(Kci,di(1,ni, (m

k)i))⊗ 1⊗s−i)×

(1⊗i ⊗ ctnv(Kci+1,di+1

(1,ni+1, (mk)i+1))⊗ 1⊗s−i−1

)=∑k∈Z

Ψi((k + 1)ri)Ψ∗i+1(kri+1 + 1).

The remaining equalities can be proved in an analogous manner.

For k = 0, 1, . . . , r, we can write

k = r1 + · · ·+ r`−1 + k′,

for unique 1 ≤ ` ≤ s and 0 ≤ k′ ≤ r` − 1. For all k such that k′ 6= 0, let

Ek := E`k′ , Fk := F`

k′ , Hk := H`k′ .


For all k such that ` 6= 1 and k′ = 0, let

Ek := E′`−1, Fk := F′`−1, Hk := H′`−1.

For k = 0,

E0 := E′0, F0 := F′0, H0 = H′0.

Theorem 5.20. The operators Ek, Fk and Hk, k = 0, 1, . . . , s−1, preserve the space

A and satisfy the Kac-Moody relations for slr. In particular, the mapping

Ek 7→ Ek, Fk 7→ Fk, Hk 7→ Hk,

defines a representation of slr on A and the linear map determined by [I] → I is an

isomorphism of slr-representations A 7→ F.

Proof: This follows from the commutativity of Diagram (5.11) and comparison

of Theorem 5.14 and Lemma 5.19 with Lemma 1.21.

By Theorem 5.20, we have a geometric version of slr. As before, to get a similar

geometric version of glr, we need to add operators corresponding to the loops on the

identity. This leads us to our main theorem, from which we conclude that A(0) is a

geometric version of Vbasic.

Theorem 5.21. For k = 0, 1, . . . , s− 1, and n ∈ Z− 0, the mapping

Ek 7→ Ek, Fk 7→ Fk, Hk 7→ Hk, I⊗ tn 7→ |n|s∑`=1

r`P`(nr`), I⊗1 7→s∑`=1

P`(0),

defines a representation of glr on the space A and the linear map determined by

[I] 7→ I is an isomorphism of glr-representations A → F. This isomorphism maps

A(0)→ F(0), and thus A(0) ∼= Vbasic.


Proof: From Theorem 5.20, the mapping

Ek 7→ Ek, Fk 7→ Fk, Hk 7→ Hk,

gives us a representation of slr on A. The additional mapping

I ⊗ tn 7→ |n|s∑`=1

r`P`(nr`), I ⊗ 1 7→s∑`=1

P`(0),

gives us a representation of glr on A by Lemma 1.21.

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