Shanghai, December 2015 - Osaka Universityariki/Shanghai2015talkfiles/Geck.pdf · This talk:...
Transcript of Shanghai, December 2015 - Osaka Universityariki/Shanghai2015talkfiles/Geck.pdf · This talk:...
Modular Harish-Chandra series
Meinolf Geck
Universitat Stuttgart
Shanghai, December 2015
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 1 / 13
The basic set-up
Representation theory of certain classes of finite groups.
G = G (q) finite classical group over Fq:
I General linear GLn(q),
I Unitary GUn(q),
I Symplectic Sp2n(q),
I Orthogonal SO2n+1(q), SO�2n(q).
(q = power of a prime p, ”defining characteristic”)
k algebraically closed of characteristic ` > 0.
Describe Irrk(G (q)): Classification (by combinatorial objects),
dimension formulas, (Brauer) character values, : : :
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 2 / 13
The basic set-up
Representation theory of certain classes of finite groups.
G = G (q) finite classical group over Fq:
I General linear GLn(q),
I Unitary GUn(q),
I Symplectic Sp2n(q),
I Orthogonal SO2n+1(q), SO�2n(q).
(q = power of a prime p, ”defining characteristic”)
k algebraically closed of characteristic ` > 0.
Describe Irrk(G (q)): Classification (by combinatorial objects),
dimension formulas, (Brauer) character values, : : :
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 2 / 13
The basic set-up
Representation theory of certain classes of finite groups.
G = G (q) finite classical group over Fq:
I General linear GLn(q),
I Unitary GUn(q),
I Symplectic Sp2n(q),
I Orthogonal SO2n+1(q), SO�2n(q).
(q = power of a prime p, ”defining characteristic”)
k algebraically closed of characteristic ` > 0.
Describe Irrk(G (q)): Classification (by combinatorial objects),
dimension formulas, (Brauer) character values, : : :
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 2 / 13
The basic set-up
Representation theory of certain classes of finite groups.
G = G (q) finite classical group over Fq:
I General linear GLn(q),
I Unitary GUn(q),
I Symplectic Sp2n(q),
I Orthogonal SO2n+1(q), SO�2n(q).
(q = power of a prime p, ”defining characteristic”)
k algebraically closed of characteristic ` > 0.
Describe Irrk(G (q)): Classification (by combinatorial objects),
dimension formulas, (Brauer) character values, : : :
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 2 / 13
The basic set-up
Representation theory of certain classes of finite groups.
G = G (q) finite classical group over Fq:
I General linear GLn(q),
I Unitary GUn(q),
I Symplectic Sp2n(q),
I Orthogonal SO2n+1(q), SO�2n(q).
(q = power of a prime p, ”defining characteristic”)
k algebraically closed of characteristic ` > 0.
Describe Irrk(G (q)):
Classification (by combinatorial objects),
dimension formulas, (Brauer) character values, : : :
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 2 / 13
The basic set-up
Representation theory of certain classes of finite groups.
G = G (q) finite classical group over Fq:
I General linear GLn(q),
I Unitary GUn(q),
I Symplectic Sp2n(q),
I Orthogonal SO2n+1(q), SO�2n(q).
(q = power of a prime p, ”defining characteristic”)
k algebraically closed of characteristic ` > 0.
Describe Irrk(G (q)): Classification (by combinatorial objects),
dimension formulas, (Brauer) character values, : : :
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 2 / 13
The basic set-up
Representation theory of certain classes of finite groups.
G = G (q) finite classical group over Fq:
I General linear GLn(q),
I Unitary GUn(q),
I Symplectic Sp2n(q),
I Orthogonal SO2n+1(q), SO�2n(q).
(q = power of a prime p, ”defining characteristic”)
k algebraically closed of characteristic ` > 0.
Describe Irrk(G (q)): Classification (by combinatorial objects),
dimension formulas, (Brauer) character values, : : :
”Defining characteristic case”: p = ` > 0
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 2 / 13
The basic set-up
Representation theory of certain classes of finite groups.
G = G (q) finite classical group over Fq:
I General linear GLn(q),
I Unitary GUn(q),
I Symplectic Sp2n(q),
I Orthogonal SO2n+1(q), SO�2n(q).
(q = power of a prime p, ”defining characteristic”)
k algebraically closed of characteristic ` > 0.
Describe Irrk(G (q)): Classification (by combinatorial objects),
dimension formulas, (Brauer) character values, : : :
”Defining characteristic case”: p = ` > 0
Algebraic group techniques (highest weight theory, : : :).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 2 / 13
The basic set-up
Representation theory of certain classes of finite groups.
G = G (q) finite classical group over Fq:
I General linear GLn(q),
I Unitary GUn(q),
I Symplectic Sp2n(q),
I Orthogonal SO2n+1(q), SO�2n(q).
(q = power of a prime p, ”defining characteristic”)
k algebraically closed of characteristic ` > 0.
Describe Irrk(G (q)): Classification (by combinatorial objects),
dimension formulas, (Brauer) character values, : : :
This talk: ”Non-defining characteristic case” ` 6= p
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 2 / 13
The basic set-up
Representation theory of certain classes of finite groups.
G = G (q) finite classical group over Fq:
I General linear GLn(q),
I Unitary GUn(q),
I Symplectic Sp2n(q),
I Orthogonal SO2n+1(q), SO�2n(q).
(q = power of a prime p, ”defining characteristic”)
k algebraically closed of characteristic ` > 0.
Describe Irrk(G (q)): Classification (by combinatorial objects),
dimension formulas, (Brauer) character values, : : :
This talk: ”Non-defining characteristic case” ` 6= p
Harish-Chandra theory: Inductive approach.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 2 / 13
Induction and restriction
Classical case:
G any finite group, L � G subgroup
IndGL : kL-mod! kG -mod; ResGL : kG -mod! kL-mod.
Transitivity, Mackey formula, right adjointness
HomkG (IndGL (X );Y ) �= HomkL(X ;ResGL (Y ))
and also left adjointness.
Harish-Chandra induction and restriction. Let P � G be a subgroup
and assume there is an epimorphism P � L.
RGL;P : kL-mod! kG -mod; �RG
L;P : kG -mod! kL-mod
[first inflate to P, then [first restrict to P, then take
induce from P to G ] fixed points under ker(P�L)]
Right adjointness
HomkG (RGL;P(X );Y ) �= HomkL(X ; �RG
L;P(Y ))
and also left adjointness, if jUP j1k 6= 0 where UP := ker(P � L).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 3 / 13
Induction and restriction
Classical case: G any finite group, L � G subgroup
IndGL : kL-mod! kG -mod; ResGL : kG -mod! kL-mod.
Transitivity, Mackey formula, right adjointness
HomkG (IndGL (X );Y ) �= HomkL(X ;ResGL (Y ))
and also left adjointness.
Harish-Chandra induction and restriction. Let P � G be a subgroup
and assume there is an epimorphism P � L.
RGL;P : kL-mod! kG -mod; �RG
L;P : kG -mod! kL-mod
[first inflate to P, then [first restrict to P, then take
induce from P to G ] fixed points under ker(P�L)]
Right adjointness
HomkG (RGL;P(X );Y ) �= HomkL(X ; �RG
L;P(Y ))
and also left adjointness, if jUP j1k 6= 0 where UP := ker(P � L).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 3 / 13
Induction and restriction
Classical case: G any finite group, L � G subgroup
IndGL : kL-mod! kG -mod; ResGL : kG -mod! kL-mod.
Transitivity, Mackey formula, right adjointness
HomkG (IndGL (X );Y ) �= HomkL(X ;ResGL (Y ))
and also left adjointness.
Harish-Chandra induction and restriction. Let P � G be a subgroup
and assume there is an epimorphism P � L.
RGL;P : kL-mod! kG -mod; �RG
L;P : kG -mod! kL-mod
[first inflate to P, then [first restrict to P, then take
induce from P to G ] fixed points under ker(P�L)]
Right adjointness
HomkG (RGL;P(X );Y ) �= HomkL(X ; �RG
L;P(Y ))
and also left adjointness, if jUP j1k 6= 0 where UP := ker(P � L).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 3 / 13
Induction and restriction
Classical case: G any finite group, L � G subgroup
IndGL : kL-mod! kG -mod; ResGL : kG -mod! kL-mod.
Transitivity, Mackey formula, right adjointness
HomkG (IndGL (X );Y ) �= HomkL(X ;ResGL (Y ))
and also left adjointness.
Harish-Chandra induction and restriction. Let P � G be a subgroup
and assume there is an epimorphism P � L.
RGL;P : kL-mod! kG -mod; �RG
L;P : kG -mod! kL-mod
[first inflate to P, then [first restrict to P, then take
induce from P to G ] fixed points under ker(P�L)]
Right adjointness
HomkG (RGL;P(X );Y ) �= HomkL(X ; �RG
L;P(Y ))
and also left adjointness, if jUP j1k 6= 0 where UP := ker(P � L).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 3 / 13
Induction and restriction
Classical case: G any finite group, L � G subgroup
IndGL : kL-mod! kG -mod; ResGL : kG -mod! kL-mod.
Transitivity, Mackey formula, right adjointness
HomkG (IndGL (X );Y ) �= HomkL(X ;ResGL (Y ))
and also left adjointness.
Harish-Chandra induction and restriction.
Let P � G be a subgroup
and assume there is an epimorphism P � L.
RGL;P : kL-mod! kG -mod; �RG
L;P : kG -mod! kL-mod
[first inflate to P, then [first restrict to P, then take
induce from P to G ] fixed points under ker(P�L)]
Right adjointness
HomkG (RGL;P(X );Y ) �= HomkL(X ; �RG
L;P(Y ))
and also left adjointness, if jUP j1k 6= 0 where UP := ker(P � L).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 3 / 13
Induction and restriction
Classical case: G any finite group, L � G subgroup
IndGL : kL-mod! kG -mod; ResGL : kG -mod! kL-mod.
Transitivity, Mackey formula, right adjointness
HomkG (IndGL (X );Y ) �= HomkL(X ;ResGL (Y ))
and also left adjointness.
Harish-Chandra induction and restriction. Let P � G be a subgroup
and assume there is an epimorphism P � L.
RGL;P : kL-mod! kG -mod; �RG
L;P : kG -mod! kL-mod
[first inflate to P, then [first restrict to P, then take
induce from P to G ] fixed points under ker(P�L)]
Right adjointness
HomkG (RGL;P(X );Y ) �= HomkL(X ; �RG
L;P(Y ))
and also left adjointness, if jUP j1k 6= 0 where UP := ker(P � L).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 3 / 13
Induction and restriction
Classical case: G any finite group, L � G subgroup
IndGL : kL-mod! kG -mod; ResGL : kG -mod! kL-mod.
Transitivity, Mackey formula, right adjointness
HomkG (IndGL (X );Y ) �= HomkL(X ;ResGL (Y ))
and also left adjointness.
Harish-Chandra induction and restriction. Let P � G be a subgroup
and assume there is an epimorphism P � L.
RGL;P : kL-mod! kG -mod; �RG
L;P : kG -mod! kL-mod
[first inflate to P, then [first restrict to P, then take
induce from P to G ] fixed points under ker(P�L)]
Right adjointness
HomkG (RGL;P(X );Y ) �= HomkL(X ; �RG
L;P(Y ))
and also left adjointness, if jUP j1k 6= 0 where UP := ker(P � L).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 3 / 13
Induction and restriction
Classical case: G any finite group, L � G subgroup
IndGL : kL-mod! kG -mod; ResGL : kG -mod! kL-mod.
Transitivity, Mackey formula, right adjointness
HomkG (IndGL (X );Y ) �= HomkL(X ;ResGL (Y ))
and also left adjointness.
Harish-Chandra induction and restriction. Let P � G be a subgroup
and assume there is an epimorphism P � L.
RGL;P : kL-mod! kG -mod; �RG
L;P : kG -mod! kL-mod
[first inflate to P, then
[first restrict to P, then take
induce from P to G ]
fixed points under ker(P�L)]
Right adjointness
HomkG (RGL;P(X );Y ) �= HomkL(X ; �RG
L;P(Y ))
and also left adjointness, if jUP j1k 6= 0 where UP := ker(P � L).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 3 / 13
Induction and restriction
Classical case: G any finite group, L � G subgroup
IndGL : kL-mod! kG -mod; ResGL : kG -mod! kL-mod.
Transitivity, Mackey formula, right adjointness
HomkG (IndGL (X );Y ) �= HomkL(X ;ResGL (Y ))
and also left adjointness.
Harish-Chandra induction and restriction. Let P � G be a subgroup
and assume there is an epimorphism P � L.
RGL;P : kL-mod! kG -mod; �RG
L;P : kG -mod! kL-mod
[first inflate to P, then [first restrict to P, then take
induce from P to G ] fixed points under ker(P�L)]
Right adjointness
HomkG (RGL;P(X );Y ) �= HomkL(X ; �RG
L;P(Y ))
and also left adjointness, if jUP j1k 6= 0 where UP := ker(P � L).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 3 / 13
Induction and restriction
Classical case: G any finite group, L � G subgroup
IndGL : kL-mod! kG -mod; ResGL : kG -mod! kL-mod.
Transitivity, Mackey formula, right adjointness
HomkG (IndGL (X );Y ) �= HomkL(X ;ResGL (Y ))
and also left adjointness.
Harish-Chandra induction and restriction. Let P � G be a subgroup
and assume there is an epimorphism P � L.
RGL;P : kL-mod! kG -mod; �RG
L;P : kG -mod! kL-mod
[first inflate to P, then [first restrict to P, then take
induce from P to G ] fixed points under ker(P�L)]
Right adjointness
HomkG (RGL;P(X );Y ) �= HomkL(X ; �RG
L;P(Y ))
and also left adjointness, if jUP j1k 6= 0 where UP := ker(P � L).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 3 / 13
Induction and restriction
Classical case: G any finite group, L � G subgroup
IndGL : kL-mod! kG -mod; ResGL : kG -mod! kL-mod.
Transitivity, Mackey formula, right adjointness
HomkG (IndGL (X );Y ) �= HomkL(X ;ResGL (Y ))
and also left adjointness.
Harish-Chandra induction and restriction. Let P � G be a subgroup
and assume there is an epimorphism P � L.
RGL;P : kL-mod! kG -mod; �RG
L;P : kG -mod! kL-mod
[first inflate to P, then [first restrict to P, then take
induce from P to G ] fixed points under ker(P�L)]
Right adjointness
HomkG (RGL;P(X );Y ) �= HomkL(X ; �RG
L;P(Y ))
and also left adjointness, if jUP j1k 6= 0 where UP := ker(P � L).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 3 / 13
General Harish-Chandra theory
Inductive setting: “`-modular Mackey systems”
LG set of pairs (L;P) where G � P � L such that jUP j1k 6= 0.
(Invariance under G -conjugation, stable under shifted intersections, etc.)
(M ;Q) � (L;P)def() UP � UQ � Q � P .
(L;P) 2 LG LG induces `-modular Mackey system LL,
with pairs (M; (Q \ P)UP=UP) where (M;Q) � (L;P) in LG .
Then HC-induction/restriction satisfy transitivity, Mackey formula,
adjointness. [Dipper–Du 1993, book by Canabes–Enguehard 2004]
Example. G = G (q) classical group (or any finite group of Lie type)
G has a BN-pair family of parabolic subgroups P � G :
LG = f(L;P) j P parabolic, L = P=UPg;
char(k) = ` 6= p
(UP = largest normal p-subgroup of P)
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 4 / 13
General Harish-Chandra theory
Inductive setting: “`-modular Mackey systems”
LG set of pairs (L;P) where G � P � L such that jUP j1k 6= 0.
(Invariance under G -conjugation, stable under shifted intersections, etc.)
(M ;Q) � (L;P)def() UP � UQ � Q � P .
(L;P) 2 LG LG induces `-modular Mackey system LL,
with pairs (M; (Q \ P)UP=UP) where (M;Q) � (L;P) in LG .
Then HC-induction/restriction satisfy transitivity, Mackey formula,
adjointness. [Dipper–Du 1993, book by Canabes–Enguehard 2004]
Example. G = G (q) classical group (or any finite group of Lie type)
G has a BN-pair family of parabolic subgroups P � G :
LG = f(L;P) j P parabolic, L = P=UPg;
char(k) = ` 6= p
(UP = largest normal p-subgroup of P)
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 4 / 13
General Harish-Chandra theory
Inductive setting: “`-modular Mackey systems”
LG set of pairs (L;P) where G � P � L such that jUP j1k 6= 0.
(Invariance under G -conjugation, stable under shifted intersections, etc.)
(M ;Q) � (L;P)def() UP � UQ � Q � P .
(L;P) 2 LG LG induces `-modular Mackey system LL,
with pairs (M; (Q \ P)UP=UP) where (M;Q) � (L;P) in LG .
Then HC-induction/restriction satisfy transitivity, Mackey formula,
adjointness. [Dipper–Du 1993, book by Canabes–Enguehard 2004]
Example. G = G (q) classical group (or any finite group of Lie type)
G has a BN-pair family of parabolic subgroups P � G :
LG = f(L;P) j P parabolic, L = P=UPg;
char(k) = ` 6= p
(UP = largest normal p-subgroup of P)
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 4 / 13
General Harish-Chandra theory
Inductive setting: “`-modular Mackey systems”
LG set of pairs (L;P) where G � P � L such that jUP j1k 6= 0.
(Invariance under G -conjugation, stable under shifted intersections, etc.)
(M ;Q) � (L;P)
def() UP � UQ � Q � P .
(L;P) 2 LG LG induces `-modular Mackey system LL,
with pairs (M; (Q \ P)UP=UP) where (M;Q) � (L;P) in LG .
Then HC-induction/restriction satisfy transitivity, Mackey formula,
adjointness. [Dipper–Du 1993, book by Canabes–Enguehard 2004]
Example. G = G (q) classical group (or any finite group of Lie type)
G has a BN-pair family of parabolic subgroups P � G :
LG = f(L;P) j P parabolic, L = P=UPg;
char(k) = ` 6= p
(UP = largest normal p-subgroup of P)
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 4 / 13
General Harish-Chandra theory
Inductive setting: “`-modular Mackey systems”
LG set of pairs (L;P) where G � P � L such that jUP j1k 6= 0.
(Invariance under G -conjugation, stable under shifted intersections, etc.)
(M ;Q) � (L;P)def() UP � UQ � Q � P .
(L;P) 2 LG LG induces `-modular Mackey system LL,
with pairs (M; (Q \ P)UP=UP) where (M;Q) � (L;P) in LG .
Then HC-induction/restriction satisfy transitivity, Mackey formula,
adjointness. [Dipper–Du 1993, book by Canabes–Enguehard 2004]
Example. G = G (q) classical group (or any finite group of Lie type)
G has a BN-pair family of parabolic subgroups P � G :
LG = f(L;P) j P parabolic, L = P=UPg;
char(k) = ` 6= p
(UP = largest normal p-subgroup of P)
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 4 / 13
General Harish-Chandra theory
Inductive setting: “`-modular Mackey systems”
LG set of pairs (L;P) where G � P � L such that jUP j1k 6= 0.
(Invariance under G -conjugation, stable under shifted intersections, etc.)
(M ;Q) � (L;P)def() UP � UQ � Q � P .
(L;P) 2 LG LG induces `-modular Mackey system LL,
with pairs (M; (Q \ P)UP=UP) where (M;Q) � (L;P) in LG .
Then HC-induction/restriction satisfy transitivity, Mackey formula,
adjointness. [Dipper–Du 1993, book by Canabes–Enguehard 2004]
Example. G = G (q) classical group (or any finite group of Lie type)
G has a BN-pair family of parabolic subgroups P � G :
LG = f(L;P) j P parabolic, L = P=UPg;
char(k) = ` 6= p
(UP = largest normal p-subgroup of P)
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 4 / 13
General Harish-Chandra theory
Inductive setting: “`-modular Mackey systems”
LG set of pairs (L;P) where G � P � L such that jUP j1k 6= 0.
(Invariance under G -conjugation, stable under shifted intersections, etc.)
(M ;Q) � (L;P)def() UP � UQ � Q � P .
(L;P) 2 LG LG induces `-modular Mackey system LL,
with pairs (M; (Q \ P)UP=UP) where (M;Q) � (L;P) in LG .
Then HC-induction/restriction satisfy transitivity, Mackey formula,
adjointness. [Dipper–Du 1993, book by Canabes–Enguehard 2004]
Example. G = G (q) classical group (or any finite group of Lie type)
G has a BN-pair family of parabolic subgroups P � G :
LG = f(L;P) j P parabolic, L = P=UPg;
char(k) = ` 6= p
(UP = largest normal p-subgroup of P)
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 4 / 13
General Harish-Chandra theory
Inductive setting: “`-modular Mackey systems”
LG set of pairs (L;P) where G � P � L such that jUP j1k 6= 0.
(Invariance under G -conjugation, stable under shifted intersections, etc.)
(M ;Q) � (L;P)def() UP � UQ � Q � P .
(L;P) 2 LG LG induces `-modular Mackey system LL,
with pairs (M; (Q \ P)UP=UP) where (M;Q) � (L;P) in LG .
Then HC-induction/restriction satisfy transitivity, Mackey formula,
adjointness. [Dipper–Du 1993, book by Canabes–Enguehard 2004]
Example. G = G (q) classical group (or any finite group of Lie type)
G has a BN-pair family of parabolic subgroups P � G :
LG = f(L;P) j P parabolic, L = P=UPg;
char(k) = ` 6= p
(UP = largest normal p-subgroup of P)
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 4 / 13
General Harish-Chandra theory
Inductive setting: “`-modular Mackey systems”
LG set of pairs (L;P) where G � P � L such that jUP j1k 6= 0.
(Invariance under G -conjugation, stable under shifted intersections, etc.)
(M ;Q) � (L;P)def() UP � UQ � Q � P .
(L;P) 2 LG LG induces `-modular Mackey system LL,
with pairs (M; (Q \ P)UP=UP) where (M;Q) � (L;P) in LG .
Then HC-induction/restriction satisfy transitivity, Mackey formula,
adjointness. [Dipper–Du 1993, book by Canabes–Enguehard 2004]
Example. G = G (q) classical group (or any finite group of Lie type)
G has a BN-pair family of parabolic subgroups P � G :
LG = f(L;P) j P parabolic, L = P=UPg;
char(k) = ` 6= p
(UP = largest normal p-subgroup of P)
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 4 / 13
General Harish-Chandra theory
Inductive setting: “`-modular Mackey systems”
LG set of pairs (L;P) where G � P � L such that jUP j1k 6= 0.
(Invariance under G -conjugation, stable under shifted intersections, etc.)
(M ;Q) � (L;P)def() UP � UQ � Q � P .
(L;P) 2 LG LG induces `-modular Mackey system LL,
with pairs (M; (Q \ P)UP=UP) where (M;Q) � (L;P) in LG .
Then HC-induction/restriction satisfy transitivity, Mackey formula,
adjointness. [Dipper–Du 1993, book by Canabes–Enguehard 2004]
Example. G = G (q) classical group (or any finite group of Lie type)
G has a BN-pair
family of parabolic subgroups P � G :
LG = f(L;P) j P parabolic, L = P=UPg;
char(k) = ` 6= p
(UP = largest normal p-subgroup of P)
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 4 / 13
General Harish-Chandra theory
Inductive setting: “`-modular Mackey systems”
LG set of pairs (L;P) where G � P � L such that jUP j1k 6= 0.
(Invariance under G -conjugation, stable under shifted intersections, etc.)
(M ;Q) � (L;P)def() UP � UQ � Q � P .
(L;P) 2 LG LG induces `-modular Mackey system LL,
with pairs (M; (Q \ P)UP=UP) where (M;Q) � (L;P) in LG .
Then HC-induction/restriction satisfy transitivity, Mackey formula,
adjointness. [Dipper–Du 1993, book by Canabes–Enguehard 2004]
Example. G = G (q) classical group (or any finite group of Lie type)
G has a BN-pair family of parabolic subgroups P � G :
LG = f(L;P) j P parabolic, L = P=UPg;
char(k) = ` 6= p
(UP = largest normal p-subgroup of P)
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 4 / 13
General Harish-Chandra theory
Inductive setting: “`-modular Mackey systems”
LG set of pairs (L;P) where G � P � L such that jUP j1k 6= 0.
(Invariance under G -conjugation, stable under shifted intersections, etc.)
(M ;Q) � (L;P)def() UP � UQ � Q � P .
(L;P) 2 LG LG induces `-modular Mackey system LL,
with pairs (M; (Q \ P)UP=UP) where (M;Q) � (L;P) in LG .
Then HC-induction/restriction satisfy transitivity, Mackey formula,
adjointness. [Dipper–Du 1993, book by Canabes–Enguehard 2004]
Example. G = G (q) classical group (or any finite group of Lie type)
G has a BN-pair family of parabolic subgroups P � G :
LG = f(L;P) j P parabolic, L = P=UPg;
char(k) = ` 6= p
(UP = largest normal p-subgroup of P)
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 4 / 13
General Harish-Chandra theory
Inductive setting: “`-modular Mackey systems”
LG set of pairs (L;P) where G � P � L such that jUP j1k 6= 0.
(Invariance under G -conjugation, stable under shifted intersections, etc.)
(M ;Q) � (L;P)def() UP � UQ � Q � P .
(L;P) 2 LG LG induces `-modular Mackey system LL,
with pairs (M; (Q \ P)UP=UP) where (M;Q) � (L;P) in LG .
Then HC-induction/restriction satisfy transitivity, Mackey formula,
adjointness. [Dipper–Du 1993, book by Canabes–Enguehard 2004]
Example. G = G (q) classical group (or any finite group of Lie type)
G has a BN-pair family of parabolic subgroups P � G :
LG = f(L;P) j P parabolic, L = P=UPg; char(k) = ` 6= p
(UP = largest normal p-subgroup of P)
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 4 / 13
General Harish-Chandra theory
Y 2 Irrk(G )
is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG and LL-cuspidal X 2 Irrk(L) Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j Y � RG
L;P(X )o
and corresponding Hecke algebra H(L;X ) := EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG , then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG and LL-cuspidal X 2 Irrk(L) Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j Y � RG
L;P(X )o
and corresponding Hecke algebra H(L;X ) := EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG , then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG
and LL-cuspidal X 2 Irrk(L) Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j Y � RG
L;P(X )o
and corresponding Hecke algebra H(L;X ) := EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG , then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG and LL-cuspidal X 2 Irrk(L)
Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j Y � RG
L;P(X )o
and corresponding Hecke algebra H(L;X ) := EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG , then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG and LL-cuspidal X 2 Irrk(L) Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j
Y � RGL;P(X )
o
and corresponding Hecke algebra H(L;X ) := EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG , then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG and LL-cuspidal X 2 Irrk(L) Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j Y � RG
L;P(X )o
and corresponding Hecke algebra H(L;X ) := EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG , then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG and LL-cuspidal X 2 Irrk(L) Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j Y � RG
L;P(X )o
and corresponding Hecke algebra H(L;X )
:= EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG , then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG and LL-cuspidal X 2 Irrk(L) Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j Y � RG
L;P(X )o
and corresponding Hecke algebra H(L;X ) := EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG , then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG and LL-cuspidal X 2 Irrk(L) Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j Y � RG
L;P(X )o
and corresponding Hecke algebra H(L;X ) := EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG , then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG and LL-cuspidal X 2 Irrk(L) Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j Y � RG
L;P(X )o
and corresponding Hecke algebra H(L;X ) := EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG ,
then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG and LL-cuspidal X 2 Irrk(L) Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j Y � RG
L;P(X )o
and corresponding Hecke algebra H(L;X ) := EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG , then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG and LL-cuspidal X 2 Irrk(L) Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j Y � RG
L;P(X )o
and corresponding Hecke algebra H(L;X ) := EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG , then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG and LL-cuspidal X 2 Irrk(L) Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j Y � RG
L;P(X )o
and corresponding Hecke algebra H(L;X ) := EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG , then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG and LL-cuspidal X 2 Irrk(L) Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j Y � RG
L;P(X )o
and corresponding Hecke algebra H(L;X ) := EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG , then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG and LL-cuspidal X 2 Irrk(L) Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j Y � RG
L;P(X )o
and corresponding Hecke algebra H(L;X ) := EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG , then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG and LL-cuspidal X 2 Irrk(L) Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j Y � RG
L;P(X )o
and corresponding Hecke algebra H(L;X ) := EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG , then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .
(L;P) 2 LG and LL-cuspidal X 2 Irrk(L) Harish-Chandra series
Irrk(G j (L;X )) :=nY 2 Irrk(G ) j Y � RG
L;P(X )o
and corresponding Hecke algebra H(L;X ) := EndkG (RGL;P(X )).
Theorem. Assume G = G (q) finite group of Lie type, LG as above.
If (L;P); (L;P 0) 2 LG , then RGL;P(X ) �= RG
L;P 0(X ) for all X ;
so just write RGL (X ): [Dipper–Du 1993; Howlett–Lehrer 1994]
Partition into Harish-Chandra series [Hiss 1993]
Irrk(G ) =a
(L;X )=�
Irrk(G j (L;X )):
Hom functor bijection [G.–Hiss–Malle 1993, G.–Hiss 1997]
Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).
[Above references for general case char(k) = ` 6= p; for ` = 0 see book by Curtis–Reiner]
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 5 / 13
General Harish-Chandra theory
Complete results
(classification of all LG -cuspidal simple modules and
description of HC-series) known for:
char(k) = 0 [Green 1955 for GLn(q); Lusztig 1980s for all others]
Explicit dimension formulas for all simple modules in this case.
GLn(q), any ` > 0, ` - q. [Dipper’s survey J. Alg. 209, 1998]
I Cuspidal simple module in char. 0 remains irreducible modulo `;
I every cuspidal simple module in char. ` arises in this way.
G (q) = unitary, symplectic or orthogonal, ` “linear prime”
[G.–Hiss–Malle 1996, Gruber–Hiss 1997]
I Numerical condition: ` - q�m�1 + 1 for all m > 1 (� = 2; 1; 1).
I Similar properties of cuspidal modules as above for GLn(q).
If ` > 0, explicit dimension formulas for simple modules not known!
In above cases, there exists a finite set of polynomials D � Q[X ]
such that dimV 2 ff (q) j f 2 Dg for all V 2 Irrk(G (q)) and all q.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 6 / 13
General Harish-Chandra theory
Complete results (classification of all LG -cuspidal simple modules and
description of HC-series) known for:
char(k) = 0 [Green 1955 for GLn(q); Lusztig 1980s for all others]
Explicit dimension formulas for all simple modules in this case.
GLn(q), any ` > 0, ` - q. [Dipper’s survey J. Alg. 209, 1998]
I Cuspidal simple module in char. 0 remains irreducible modulo `;
I every cuspidal simple module in char. ` arises in this way.
G (q) = unitary, symplectic or orthogonal, ` “linear prime”
[G.–Hiss–Malle 1996, Gruber–Hiss 1997]
I Numerical condition: ` - q�m�1 + 1 for all m > 1 (� = 2; 1; 1).
I Similar properties of cuspidal modules as above for GLn(q).
If ` > 0, explicit dimension formulas for simple modules not known!
In above cases, there exists a finite set of polynomials D � Q[X ]
such that dimV 2 ff (q) j f 2 Dg for all V 2 Irrk(G (q)) and all q.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 6 / 13
General Harish-Chandra theory
Complete results (classification of all LG -cuspidal simple modules and
description of HC-series) known for:
char(k) = 0 [Green 1955 for GLn(q); Lusztig 1980s for all others]
Explicit dimension formulas for all simple modules in this case.
GLn(q), any ` > 0, ` - q. [Dipper’s survey J. Alg. 209, 1998]
I Cuspidal simple module in char. 0 remains irreducible modulo `;
I every cuspidal simple module in char. ` arises in this way.
G (q) = unitary, symplectic or orthogonal, ` “linear prime”
[G.–Hiss–Malle 1996, Gruber–Hiss 1997]
I Numerical condition: ` - q�m�1 + 1 for all m > 1 (� = 2; 1; 1).
I Similar properties of cuspidal modules as above for GLn(q).
If ` > 0, explicit dimension formulas for simple modules not known!
In above cases, there exists a finite set of polynomials D � Q[X ]
such that dimV 2 ff (q) j f 2 Dg for all V 2 Irrk(G (q)) and all q.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 6 / 13
General Harish-Chandra theory
Complete results (classification of all LG -cuspidal simple modules and
description of HC-series) known for:
char(k) = 0 [Green 1955 for GLn(q); Lusztig 1980s for all others]
Explicit dimension formulas for all simple modules in this case.
GLn(q), any ` > 0, ` - q. [Dipper’s survey J. Alg. 209, 1998]
I Cuspidal simple module in char. 0 remains irreducible modulo `;
I every cuspidal simple module in char. ` arises in this way.
G (q) = unitary, symplectic or orthogonal, ` “linear prime”
[G.–Hiss–Malle 1996, Gruber–Hiss 1997]
I Numerical condition: ` - q�m�1 + 1 for all m > 1 (� = 2; 1; 1).
I Similar properties of cuspidal modules as above for GLn(q).
If ` > 0, explicit dimension formulas for simple modules not known!
In above cases, there exists a finite set of polynomials D � Q[X ]
such that dimV 2 ff (q) j f 2 Dg for all V 2 Irrk(G (q)) and all q.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 6 / 13
General Harish-Chandra theory
Complete results (classification of all LG -cuspidal simple modules and
description of HC-series) known for:
char(k) = 0 [Green 1955 for GLn(q); Lusztig 1980s for all others]
Explicit dimension formulas for all simple modules in this case.
GLn(q), any ` > 0, ` - q. [Dipper’s survey J. Alg. 209, 1998]
I Cuspidal simple module in char. 0 remains irreducible modulo `;
I every cuspidal simple module in char. ` arises in this way.
G (q) = unitary, symplectic or orthogonal, ` “linear prime”
[G.–Hiss–Malle 1996, Gruber–Hiss 1997]
I Numerical condition: ` - q�m�1 + 1 for all m > 1 (� = 2; 1; 1).
I Similar properties of cuspidal modules as above for GLn(q).
If ` > 0, explicit dimension formulas for simple modules not known!
In above cases, there exists a finite set of polynomials D � Q[X ]
such that dimV 2 ff (q) j f 2 Dg for all V 2 Irrk(G (q)) and all q.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 6 / 13
General Harish-Chandra theory
Complete results (classification of all LG -cuspidal simple modules and
description of HC-series) known for:
char(k) = 0 [Green 1955 for GLn(q); Lusztig 1980s for all others]
Explicit dimension formulas for all simple modules in this case.
GLn(q), any ` > 0, ` - q. [Dipper’s survey J. Alg. 209, 1998]
I Cuspidal simple module in char. 0 remains irreducible modulo `;
I every cuspidal simple module in char. ` arises in this way.
G (q) = unitary, symplectic or orthogonal, ` “linear prime”
[G.–Hiss–Malle 1996, Gruber–Hiss 1997]
I Numerical condition: ` - q�m�1 + 1 for all m > 1 (� = 2; 1; 1).
I Similar properties of cuspidal modules as above for GLn(q).
If ` > 0, explicit dimension formulas for simple modules not known!
In above cases, there exists a finite set of polynomials D � Q[X ]
such that dimV 2 ff (q) j f 2 Dg for all V 2 Irrk(G (q)) and all q.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 6 / 13
General Harish-Chandra theory
Complete results (classification of all LG -cuspidal simple modules and
description of HC-series) known for:
char(k) = 0 [Green 1955 for GLn(q); Lusztig 1980s for all others]
Explicit dimension formulas for all simple modules in this case.
GLn(q), any ` > 0, ` - q. [Dipper’s survey J. Alg. 209, 1998]
I Cuspidal simple module in char. 0 remains irreducible modulo `;
I every cuspidal simple module in char. ` arises in this way.
G (q) = unitary, symplectic or orthogonal, ` “linear prime”
[G.–Hiss–Malle 1996, Gruber–Hiss 1997]
I Numerical condition: ` - q�m�1 + 1 for all m > 1 (� = 2; 1; 1).
I Similar properties of cuspidal modules as above for GLn(q).
If ` > 0, explicit dimension formulas for simple modules not known!
In above cases, there exists a finite set of polynomials D � Q[X ]
such that dimV 2 ff (q) j f 2 Dg for all V 2 Irrk(G (q)) and all q.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 6 / 13
General Harish-Chandra theory
Complete results (classification of all LG -cuspidal simple modules and
description of HC-series) known for:
char(k) = 0 [Green 1955 for GLn(q); Lusztig 1980s for all others]
Explicit dimension formulas for all simple modules in this case.
GLn(q), any ` > 0, ` - q. [Dipper’s survey J. Alg. 209, 1998]
I Cuspidal simple module in char. 0 remains irreducible modulo `;
I every cuspidal simple module in char. ` arises in this way.
G (q) = unitary, symplectic or orthogonal, ` “linear prime”
[G.–Hiss–Malle 1996, Gruber–Hiss 1997]
I Numerical condition: ` - q�m�1 + 1 for all m > 1 (� = 2; 1; 1).
I Similar properties of cuspidal modules as above for GLn(q).
If ` > 0, explicit dimension formulas for simple modules not known!
In above cases, there exists a finite set of polynomials D � Q[X ]
such that dimV 2 ff (q) j f 2 Dg for all V 2 Irrk(G (q)) and all q.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 6 / 13
General Harish-Chandra theory
Complete results (classification of all LG -cuspidal simple modules and
description of HC-series) known for:
char(k) = 0 [Green 1955 for GLn(q); Lusztig 1980s for all others]
Explicit dimension formulas for all simple modules in this case.
GLn(q), any ` > 0, ` - q. [Dipper’s survey J. Alg. 209, 1998]
I Cuspidal simple module in char. 0 remains irreducible modulo `;
I every cuspidal simple module in char. ` arises in this way.
G (q) = unitary, symplectic or orthogonal, ` “linear prime”
[G.–Hiss–Malle 1996, Gruber–Hiss 1997]
I Numerical condition: ` - q�m�1 + 1 for all m > 1 (� = 2; 1; 1).
I Similar properties of cuspidal modules as above for GLn(q).
If ` > 0, explicit dimension formulas for simple modules not known!
In above cases, there exists a finite set of polynomials D � Q[X ]
such that dimV 2 ff (q) j f 2 Dg for all V 2 Irrk(G (q)) and all q.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 6 / 13
General Harish-Chandra theory
Complete results (classification of all LG -cuspidal simple modules and
description of HC-series) known for:
char(k) = 0 [Green 1955 for GLn(q); Lusztig 1980s for all others]
Explicit dimension formulas for all simple modules in this case.
GLn(q), any ` > 0, ` - q. [Dipper’s survey J. Alg. 209, 1998]
I Cuspidal simple module in char. 0 remains irreducible modulo `;
I every cuspidal simple module in char. ` arises in this way.
G (q) = unitary, symplectic or orthogonal, ` “linear prime”
[G.–Hiss–Malle 1996, Gruber–Hiss 1997]
I Numerical condition: ` - q�m�1 + 1 for all m > 1 (� = 2; 1; 1).
I Similar properties of cuspidal modules as above for GLn(q).
If ` > 0, explicit dimension formulas for simple modules not known!
In above cases, there exists a finite set of polynomials D � Q[X ]
such that dimV 2 ff (q) j f 2 Dg for all V 2 Irrk(G (q)) and all q.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 6 / 13
General Harish-Chandra theory
Complete results (classification of all LG -cuspidal simple modules and
description of HC-series) known for:
char(k) = 0 [Green 1955 for GLn(q); Lusztig 1980s for all others]
Explicit dimension formulas for all simple modules in this case.
GLn(q), any ` > 0, ` - q. [Dipper’s survey J. Alg. 209, 1998]
I Cuspidal simple module in char. 0 remains irreducible modulo `;
I every cuspidal simple module in char. ` arises in this way.
G (q) = unitary, symplectic or orthogonal, ` “linear prime”
[G.–Hiss–Malle 1996, Gruber–Hiss 1997]
I Numerical condition: ` - q�m�1 + 1 for all m > 1 (� = 2; 1; 1).
I Similar properties of cuspidal modules as above for GLn(q).
If ` > 0, explicit dimension formulas for simple modules not known!
In above cases, there exists a finite set of polynomials D � Q[X ]
such that dimV 2 ff (q) j f 2 Dg for all V 2 Irrk(G (q)) and all q.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 6 / 13
General Harish-Chandra theory
Complete results (classification of all LG -cuspidal simple modules and
description of HC-series) known for:
char(k) = 0 [Green 1955 for GLn(q); Lusztig 1980s for all others]
Explicit dimension formulas for all simple modules in this case.
GLn(q), any ` > 0, ` - q. [Dipper’s survey J. Alg. 209, 1998]
I Cuspidal simple module in char. 0 remains irreducible modulo `;
I every cuspidal simple module in char. ` arises in this way.
G (q) = unitary, symplectic or orthogonal, ` “linear prime”
[G.–Hiss–Malle 1996, Gruber–Hiss 1997]
I Numerical condition: ` - q�m�1 + 1 for all m > 1 (� = 2; 1; 1).
I Similar properties of cuspidal modules as above for GLn(q).
If ` > 0, explicit dimension formulas for simple modules not known!
In above cases, there exists a finite set of polynomials D � Q[X ]
such that dimV 2 ff (q) j f 2 Dg for all V 2 Irrk(G (q)) and all q.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 6 / 13
General Harish-Chandra theory
Complete results (classification of all LG -cuspidal simple modules and
description of HC-series) known for:
char(k) = 0 [Green 1955 for GLn(q); Lusztig 1980s for all others]
Explicit dimension formulas for all simple modules in this case.
GLn(q), any ` > 0, ` - q. [Dipper’s survey J. Alg. 209, 1998]
I Cuspidal simple module in char. 0 remains irreducible modulo `;
I every cuspidal simple module in char. ` arises in this way.
G (q) = unitary, symplectic or orthogonal, ` “linear prime”
[G.–Hiss–Malle 1996, Gruber–Hiss 1997]
I Numerical condition: ` - q�m�1 + 1 for all m > 1 (� = 2; 1; 1).
I Similar properties of cuspidal modules as above for GLn(q).
If ` > 0, explicit dimension formulas for simple modules not known!
In above cases, there exists a finite set of polynomials D � Q[X ]
such that dimV 2 ff (q) j f 2 Dg for all V 2 Irrk(G (q)) and all q.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 6 / 13
One particular case: Unipotent principal series
B � G Borel subgroup, H = B=UB maximal torus, kH LH-cuspidal.
Irrk(G j (H ; kH)) = fV 2 Irrk(G ) j FixB(V ) 6= f0gg
H := H(H ; kH) Iwahori–Hecke algebra associated with (W ; S),
standard basis fTw j w 2W g, generators fTs j s 2 Sg satisfy braid
relations plus quadratic relations (Ts � qcs )(Ts + 1) = 0 for s 2 S .
H “cellular” in the sense of Graham–Lehrer, cell modules
naturally parametrised by IrrC(W ): [G. 2007]
Irrk(G j (H ; kH))1�1 ! Irr(H) has a natural parametrisation by
a subset Λ�k � IrrC(W ). (If char(k) = 0, then Λ�
k = IrrC(W ).)
Determination of Λ�k [Jacon 2004, book G.–J. 2011]. — Based on
Lascoux–Leclerc–Thibon conjecture 1996, Ariki’s proof 2000:
Connection modular representations ! crystal graph theory.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 7 / 13
One particular case: Unipotent principal series
B � G Borel subgroup, H = B=UB maximal torus, kH LH-cuspidal.
Irrk(G j (H ; kH))
= fV 2 Irrk(G ) j FixB(V ) 6= f0gg
H := H(H ; kH) Iwahori–Hecke algebra associated with (W ; S),
standard basis fTw j w 2W g, generators fTs j s 2 Sg satisfy braid
relations plus quadratic relations (Ts � qcs )(Ts + 1) = 0 for s 2 S .
H “cellular” in the sense of Graham–Lehrer, cell modules
naturally parametrised by IrrC(W ): [G. 2007]
Irrk(G j (H ; kH))1�1 ! Irr(H) has a natural parametrisation by
a subset Λ�k � IrrC(W ). (If char(k) = 0, then Λ�
k = IrrC(W ).)
Determination of Λ�k [Jacon 2004, book G.–J. 2011]. — Based on
Lascoux–Leclerc–Thibon conjecture 1996, Ariki’s proof 2000:
Connection modular representations ! crystal graph theory.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 7 / 13
One particular case: Unipotent principal series
B � G Borel subgroup, H = B=UB maximal torus, kH LH-cuspidal.
Irrk(G j (H ; kH)) = fV 2 Irrk(G ) j FixB(V ) 6= f0gg
H := H(H ; kH) Iwahori–Hecke algebra associated with (W ; S),
standard basis fTw j w 2W g, generators fTs j s 2 Sg satisfy braid
relations plus quadratic relations (Ts � qcs )(Ts + 1) = 0 for s 2 S .
H “cellular” in the sense of Graham–Lehrer, cell modules
naturally parametrised by IrrC(W ): [G. 2007]
Irrk(G j (H ; kH))1�1 ! Irr(H) has a natural parametrisation by
a subset Λ�k � IrrC(W ). (If char(k) = 0, then Λ�
k = IrrC(W ).)
Determination of Λ�k [Jacon 2004, book G.–J. 2011]. — Based on
Lascoux–Leclerc–Thibon conjecture 1996, Ariki’s proof 2000:
Connection modular representations ! crystal graph theory.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 7 / 13
One particular case: Unipotent principal series
B � G Borel subgroup, H = B=UB maximal torus, kH LH-cuspidal.
Irrk(G j (H ; kH)) = fV 2 Irrk(G ) j FixB(V ) 6= f0gg
H := H(H ; kH) Iwahori–Hecke algebra associated with (W ; S),
standard basis fTw j w 2W g, generators fTs j s 2 Sg satisfy braid
relations plus quadratic relations (Ts � qcs )(Ts + 1) = 0 for s 2 S .
H “cellular” in the sense of Graham–Lehrer, cell modules
naturally parametrised by IrrC(W ): [G. 2007]
Irrk(G j (H ; kH))1�1 ! Irr(H) has a natural parametrisation by
a subset Λ�k � IrrC(W ). (If char(k) = 0, then Λ�
k = IrrC(W ).)
Determination of Λ�k [Jacon 2004, book G.–J. 2011]. — Based on
Lascoux–Leclerc–Thibon conjecture 1996, Ariki’s proof 2000:
Connection modular representations ! crystal graph theory.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 7 / 13
One particular case: Unipotent principal series
B � G Borel subgroup, H = B=UB maximal torus, kH LH-cuspidal.
Irrk(G j (H ; kH)) = fV 2 Irrk(G ) j FixB(V ) 6= f0gg
H := H(H ; kH) Iwahori–Hecke algebra associated with (W ; S),
standard basis fTw j w 2W g, generators fTs j s 2 Sg satisfy braid
relations plus quadratic relations (Ts � qcs )(Ts + 1) = 0 for s 2 S .
H “cellular” in the sense of Graham–Lehrer, cell modules
naturally parametrised by IrrC(W ): [G. 2007]
Irrk(G j (H ; kH))1�1 ! Irr(H) has a natural parametrisation by
a subset Λ�k � IrrC(W ). (If char(k) = 0, then Λ�
k = IrrC(W ).)
Determination of Λ�k [Jacon 2004, book G.–J. 2011]. — Based on
Lascoux–Leclerc–Thibon conjecture 1996, Ariki’s proof 2000:
Connection modular representations ! crystal graph theory.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 7 / 13
One particular case: Unipotent principal series
B � G Borel subgroup, H = B=UB maximal torus, kH LH-cuspidal.
Irrk(G j (H ; kH)) = fV 2 Irrk(G ) j FixB(V ) 6= f0gg
H := H(H ; kH) Iwahori–Hecke algebra associated with (W ; S),
standard basis fTw j w 2W g, generators fTs j s 2 Sg satisfy braid
relations plus quadratic relations (Ts � qcs )(Ts + 1) = 0 for s 2 S .
H “cellular” in the sense of Graham–Lehrer,
cell modules
naturally parametrised by IrrC(W ): [G. 2007]
Irrk(G j (H ; kH))1�1 ! Irr(H) has a natural parametrisation by
a subset Λ�k � IrrC(W ). (If char(k) = 0, then Λ�
k = IrrC(W ).)
Determination of Λ�k [Jacon 2004, book G.–J. 2011]. — Based on
Lascoux–Leclerc–Thibon conjecture 1996, Ariki’s proof 2000:
Connection modular representations ! crystal graph theory.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 7 / 13
One particular case: Unipotent principal series
B � G Borel subgroup, H = B=UB maximal torus, kH LH-cuspidal.
Irrk(G j (H ; kH)) = fV 2 Irrk(G ) j FixB(V ) 6= f0gg
H := H(H ; kH) Iwahori–Hecke algebra associated with (W ; S),
standard basis fTw j w 2W g, generators fTs j s 2 Sg satisfy braid
relations plus quadratic relations (Ts � qcs )(Ts + 1) = 0 for s 2 S .
H “cellular” in the sense of Graham–Lehrer, cell modules
naturally parametrised by IrrC(W ): [G. 2007]
Irrk(G j (H ; kH))1�1 ! Irr(H) has a natural parametrisation by
a subset Λ�k � IrrC(W ). (If char(k) = 0, then Λ�
k = IrrC(W ).)
Determination of Λ�k [Jacon 2004, book G.–J. 2011]. — Based on
Lascoux–Leclerc–Thibon conjecture 1996, Ariki’s proof 2000:
Connection modular representations ! crystal graph theory.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 7 / 13
One particular case: Unipotent principal series
B � G Borel subgroup, H = B=UB maximal torus, kH LH-cuspidal.
Irrk(G j (H ; kH)) = fV 2 Irrk(G ) j FixB(V ) 6= f0gg
H := H(H ; kH) Iwahori–Hecke algebra associated with (W ; S),
standard basis fTw j w 2W g, generators fTs j s 2 Sg satisfy braid
relations plus quadratic relations (Ts � qcs )(Ts + 1) = 0 for s 2 S .
H “cellular” in the sense of Graham–Lehrer, cell modules
naturally parametrised by IrrC(W ): [G. 2007]
Irrk(G j (H ; kH))1�1 ! Irr(H)
has a natural parametrisation by
a subset Λ�k � IrrC(W ). (If char(k) = 0, then Λ�
k = IrrC(W ).)
Determination of Λ�k [Jacon 2004, book G.–J. 2011]. — Based on
Lascoux–Leclerc–Thibon conjecture 1996, Ariki’s proof 2000:
Connection modular representations ! crystal graph theory.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 7 / 13
One particular case: Unipotent principal series
B � G Borel subgroup, H = B=UB maximal torus, kH LH-cuspidal.
Irrk(G j (H ; kH)) = fV 2 Irrk(G ) j FixB(V ) 6= f0gg
H := H(H ; kH) Iwahori–Hecke algebra associated with (W ; S),
standard basis fTw j w 2W g, generators fTs j s 2 Sg satisfy braid
relations plus quadratic relations (Ts � qcs )(Ts + 1) = 0 for s 2 S .
H “cellular” in the sense of Graham–Lehrer, cell modules
naturally parametrised by IrrC(W ): [G. 2007]
Irrk(G j (H ; kH))1�1 ! Irr(H) has a natural parametrisation by
a subset Λ�k � IrrC(W ).
(If char(k) = 0, then Λ�
k = IrrC(W ).)
Determination of Λ�k [Jacon 2004, book G.–J. 2011]. — Based on
Lascoux–Leclerc–Thibon conjecture 1996, Ariki’s proof 2000:
Connection modular representations ! crystal graph theory.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 7 / 13
One particular case: Unipotent principal series
B � G Borel subgroup, H = B=UB maximal torus, kH LH-cuspidal.
Irrk(G j (H ; kH)) = fV 2 Irrk(G ) j FixB(V ) 6= f0gg
H := H(H ; kH) Iwahori–Hecke algebra associated with (W ; S),
standard basis fTw j w 2W g, generators fTs j s 2 Sg satisfy braid
relations plus quadratic relations (Ts � qcs )(Ts + 1) = 0 for s 2 S .
H “cellular” in the sense of Graham–Lehrer, cell modules
naturally parametrised by IrrC(W ): [G. 2007]
Irrk(G j (H ; kH))1�1 ! Irr(H) has a natural parametrisation by
a subset Λ�k � IrrC(W ). (If char(k) = 0, then Λ�
k = IrrC(W ).)
Determination of Λ�k [Jacon 2004, book G.–J. 2011]. — Based on
Lascoux–Leclerc–Thibon conjecture 1996, Ariki’s proof 2000:
Connection modular representations ! crystal graph theory.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 7 / 13
One particular case: Unipotent principal series
B � G Borel subgroup, H = B=UB maximal torus, kH LH-cuspidal.
Irrk(G j (H ; kH)) = fV 2 Irrk(G ) j FixB(V ) 6= f0gg
H := H(H ; kH) Iwahori–Hecke algebra associated with (W ; S),
standard basis fTw j w 2W g, generators fTs j s 2 Sg satisfy braid
relations plus quadratic relations (Ts � qcs )(Ts + 1) = 0 for s 2 S .
H “cellular” in the sense of Graham–Lehrer, cell modules
naturally parametrised by IrrC(W ): [G. 2007]
Irrk(G j (H ; kH))1�1 ! Irr(H) has a natural parametrisation by
a subset Λ�k � IrrC(W ). (If char(k) = 0, then Λ�
k = IrrC(W ).)
Determination of Λ�k [Jacon 2004, book G.–J. 2011].
— Based on
Lascoux–Leclerc–Thibon conjecture 1996, Ariki’s proof 2000:
Connection modular representations ! crystal graph theory.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 7 / 13
One particular case: Unipotent principal series
B � G Borel subgroup, H = B=UB maximal torus, kH LH-cuspidal.
Irrk(G j (H ; kH)) = fV 2 Irrk(G ) j FixB(V ) 6= f0gg
H := H(H ; kH) Iwahori–Hecke algebra associated with (W ; S),
standard basis fTw j w 2W g, generators fTs j s 2 Sg satisfy braid
relations plus quadratic relations (Ts � qcs )(Ts + 1) = 0 for s 2 S .
H “cellular” in the sense of Graham–Lehrer, cell modules
naturally parametrised by IrrC(W ): [G. 2007]
Irrk(G j (H ; kH))1�1 ! Irr(H) has a natural parametrisation by
a subset Λ�k � IrrC(W ). (If char(k) = 0, then Λ�
k = IrrC(W ).)
Determination of Λ�k [Jacon 2004, book G.–J. 2011]. — Based on
Lascoux–Leclerc–Thibon conjecture 1996, Ariki’s proof 2000:
Connection modular representations ! crystal graph theory.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 7 / 13
One particular case: Unipotent principal series
B � G Borel subgroup, H = B=UB maximal torus, kH LH-cuspidal.
Irrk(G j (H ; kH)) = fV 2 Irrk(G ) j FixB(V ) 6= f0gg
H := H(H ; kH) Iwahori–Hecke algebra associated with (W ; S),
standard basis fTw j w 2W g, generators fTs j s 2 Sg satisfy braid
relations plus quadratic relations (Ts � qcs )(Ts + 1) = 0 for s 2 S .
H “cellular” in the sense of Graham–Lehrer, cell modules
naturally parametrised by IrrC(W ): [G. 2007]
Irrk(G j (H ; kH))1�1 ! Irr(H) has a natural parametrisation by
a subset Λ�k � IrrC(W ). (If char(k) = 0, then Λ�
k = IrrC(W ).)
Determination of Λ�k [Jacon 2004, book G.–J. 2011]. — Based on
Lascoux–Leclerc–Thibon conjecture 1996, Ariki’s proof 2000:
Connection modular representations ! crystal graph theory.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 7 / 13
One particular case: Unipotent principal series
Example.
H has two 1-dimensional representations
ind : Ts 7! qcs and " : Ts 7! �1 (s 2 S).
Easy to see: kG $ ind under Irrk(G j (H ; kH))1�1 ! Irr(H).
StG = Steinberg module. Simple , [G : B]1k 6= 0. [Steinberg 1957]
soc(StG ) always simple [Tinberg 1986].
soc(Stk)$ " under Irrk(G j (H ; kH))1�1 ! Irr(H): [G. 2015]
Recall: Natural parametrisation of Irr(H) by Λ�k � IrrC(W ).
What is the label of soc(StG )$ " in Λ�k ?
(If [G : B]1k 6= 0, then Λ�
k = IrrC(W ), label is sign representation of W .)
If G = GLn(q), then W �= Sn and IrrC(W )$ fpartitions of ng.
Label of " corresponds to image of (n) under Mullineux involution.
Analogous result for classical groups G (q). Description of Mullineux
involution using crystal graph theory [Jacon arXiv:1509:03417].
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 8 / 13
One particular case: Unipotent principal series
Example. H has two 1-dimensional representations
ind : Ts 7! qcs
and " : Ts 7! �1 (s 2 S).
Easy to see: kG $ ind under Irrk(G j (H ; kH))1�1 ! Irr(H).
StG = Steinberg module. Simple , [G : B]1k 6= 0. [Steinberg 1957]
soc(StG ) always simple [Tinberg 1986].
soc(Stk)$ " under Irrk(G j (H ; kH))1�1 ! Irr(H): [G. 2015]
Recall: Natural parametrisation of Irr(H) by Λ�k � IrrC(W ).
What is the label of soc(StG )$ " in Λ�k ?
(If [G : B]1k 6= 0, then Λ�
k = IrrC(W ), label is sign representation of W .)
If G = GLn(q), then W �= Sn and IrrC(W )$ fpartitions of ng.
Label of " corresponds to image of (n) under Mullineux involution.
Analogous result for classical groups G (q). Description of Mullineux
involution using crystal graph theory [Jacon arXiv:1509:03417].
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 8 / 13
One particular case: Unipotent principal series
Example. H has two 1-dimensional representations
ind : Ts 7! qcs and " : Ts 7! �1 (s 2 S).
Easy to see: kG $ ind under Irrk(G j (H ; kH))1�1 ! Irr(H).
StG = Steinberg module. Simple , [G : B]1k 6= 0. [Steinberg 1957]
soc(StG ) always simple [Tinberg 1986].
soc(Stk)$ " under Irrk(G j (H ; kH))1�1 ! Irr(H): [G. 2015]
Recall: Natural parametrisation of Irr(H) by Λ�k � IrrC(W ).
What is the label of soc(StG )$ " in Λ�k ?
(If [G : B]1k 6= 0, then Λ�
k = IrrC(W ), label is sign representation of W .)
If G = GLn(q), then W �= Sn and IrrC(W )$ fpartitions of ng.
Label of " corresponds to image of (n) under Mullineux involution.
Analogous result for classical groups G (q). Description of Mullineux
involution using crystal graph theory [Jacon arXiv:1509:03417].
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 8 / 13
One particular case: Unipotent principal series
Example. H has two 1-dimensional representations
ind : Ts 7! qcs and " : Ts 7! �1 (s 2 S).
Easy to see: kG $ ind under Irrk(G j (H ; kH))1�1 ! Irr(H).
StG = Steinberg module. Simple , [G : B]1k 6= 0. [Steinberg 1957]
soc(StG ) always simple [Tinberg 1986].
soc(Stk)$ " under Irrk(G j (H ; kH))1�1 ! Irr(H): [G. 2015]
Recall: Natural parametrisation of Irr(H) by Λ�k � IrrC(W ).
What is the label of soc(StG )$ " in Λ�k ?
(If [G : B]1k 6= 0, then Λ�
k = IrrC(W ), label is sign representation of W .)
If G = GLn(q), then W �= Sn and IrrC(W )$ fpartitions of ng.
Label of " corresponds to image of (n) under Mullineux involution.
Analogous result for classical groups G (q). Description of Mullineux
involution using crystal graph theory [Jacon arXiv:1509:03417].
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 8 / 13
One particular case: Unipotent principal series
Example. H has two 1-dimensional representations
ind : Ts 7! qcs and " : Ts 7! �1 (s 2 S).
Easy to see: kG $ ind under Irrk(G j (H ; kH))1�1 ! Irr(H).
StG = Steinberg module.
Simple , [G : B]1k 6= 0. [Steinberg 1957]
soc(StG ) always simple [Tinberg 1986].
soc(Stk)$ " under Irrk(G j (H ; kH))1�1 ! Irr(H): [G. 2015]
Recall: Natural parametrisation of Irr(H) by Λ�k � IrrC(W ).
What is the label of soc(StG )$ " in Λ�k ?
(If [G : B]1k 6= 0, then Λ�
k = IrrC(W ), label is sign representation of W .)
If G = GLn(q), then W �= Sn and IrrC(W )$ fpartitions of ng.
Label of " corresponds to image of (n) under Mullineux involution.
Analogous result for classical groups G (q). Description of Mullineux
involution using crystal graph theory [Jacon arXiv:1509:03417].
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 8 / 13
One particular case: Unipotent principal series
Example. H has two 1-dimensional representations
ind : Ts 7! qcs and " : Ts 7! �1 (s 2 S).
Easy to see: kG $ ind under Irrk(G j (H ; kH))1�1 ! Irr(H).
StG = Steinberg module. Simple , [G : B]1k 6= 0. [Steinberg 1957]
soc(StG ) always simple [Tinberg 1986].
soc(Stk)$ " under Irrk(G j (H ; kH))1�1 ! Irr(H): [G. 2015]
Recall: Natural parametrisation of Irr(H) by Λ�k � IrrC(W ).
What is the label of soc(StG )$ " in Λ�k ?
(If [G : B]1k 6= 0, then Λ�
k = IrrC(W ), label is sign representation of W .)
If G = GLn(q), then W �= Sn and IrrC(W )$ fpartitions of ng.
Label of " corresponds to image of (n) under Mullineux involution.
Analogous result for classical groups G (q). Description of Mullineux
involution using crystal graph theory [Jacon arXiv:1509:03417].
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 8 / 13
One particular case: Unipotent principal series
Example. H has two 1-dimensional representations
ind : Ts 7! qcs and " : Ts 7! �1 (s 2 S).
Easy to see: kG $ ind under Irrk(G j (H ; kH))1�1 ! Irr(H).
StG = Steinberg module. Simple , [G : B]1k 6= 0. [Steinberg 1957]
soc(StG ) always simple [Tinberg 1986].
soc(Stk)$ " under Irrk(G j (H ; kH))1�1 ! Irr(H): [G. 2015]
Recall: Natural parametrisation of Irr(H) by Λ�k � IrrC(W ).
What is the label of soc(StG )$ " in Λ�k ?
(If [G : B]1k 6= 0, then Λ�
k = IrrC(W ), label is sign representation of W .)
If G = GLn(q), then W �= Sn and IrrC(W )$ fpartitions of ng.
Label of " corresponds to image of (n) under Mullineux involution.
Analogous result for classical groups G (q). Description of Mullineux
involution using crystal graph theory [Jacon arXiv:1509:03417].
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 8 / 13
One particular case: Unipotent principal series
Example. H has two 1-dimensional representations
ind : Ts 7! qcs and " : Ts 7! �1 (s 2 S).
Easy to see: kG $ ind under Irrk(G j (H ; kH))1�1 ! Irr(H).
StG = Steinberg module. Simple , [G : B]1k 6= 0. [Steinberg 1957]
soc(StG ) always simple [Tinberg 1986].
soc(Stk)$ " under Irrk(G j (H ; kH))1�1 ! Irr(H): [G. 2015]
Recall: Natural parametrisation of Irr(H) by Λ�k � IrrC(W ).
What is the label of soc(StG )$ " in Λ�k ?
(If [G : B]1k 6= 0, then Λ�
k = IrrC(W ), label is sign representation of W .)
If G = GLn(q), then W �= Sn and IrrC(W )$ fpartitions of ng.
Label of " corresponds to image of (n) under Mullineux involution.
Analogous result for classical groups G (q). Description of Mullineux
involution using crystal graph theory [Jacon arXiv:1509:03417].
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 8 / 13
One particular case: Unipotent principal series
Example. H has two 1-dimensional representations
ind : Ts 7! qcs and " : Ts 7! �1 (s 2 S).
Easy to see: kG $ ind under Irrk(G j (H ; kH))1�1 ! Irr(H).
StG = Steinberg module. Simple , [G : B]1k 6= 0. [Steinberg 1957]
soc(StG ) always simple [Tinberg 1986].
soc(Stk)$ " under Irrk(G j (H ; kH))1�1 ! Irr(H): [G. 2015]
Recall: Natural parametrisation of Irr(H) by Λ�k � IrrC(W ).
What is the label of soc(StG )$ " in Λ�k ?
(If [G : B]1k 6= 0, then Λ�
k = IrrC(W ), label is sign representation of W .)
If G = GLn(q), then W �= Sn and IrrC(W )$ fpartitions of ng.
Label of " corresponds to image of (n) under Mullineux involution.
Analogous result for classical groups G (q). Description of Mullineux
involution using crystal graph theory [Jacon arXiv:1509:03417].
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 8 / 13
One particular case: Unipotent principal series
Example. H has two 1-dimensional representations
ind : Ts 7! qcs and " : Ts 7! �1 (s 2 S).
Easy to see: kG $ ind under Irrk(G j (H ; kH))1�1 ! Irr(H).
StG = Steinberg module. Simple , [G : B]1k 6= 0. [Steinberg 1957]
soc(StG ) always simple [Tinberg 1986].
soc(Stk)$ " under Irrk(G j (H ; kH))1�1 ! Irr(H): [G. 2015]
Recall: Natural parametrisation of Irr(H) by Λ�k � IrrC(W ).
What is the label of soc(StG )$ " in Λ�k ?
(If [G : B]1k 6= 0, then Λ�
k = IrrC(W ), label is sign representation of W .)
If G = GLn(q), then W �= Sn and IrrC(W )$ fpartitions of ng.
Label of " corresponds to image of (n) under Mullineux involution.
Analogous result for classical groups G (q). Description of Mullineux
involution using crystal graph theory [Jacon arXiv:1509:03417].
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 8 / 13
One particular case: Unipotent principal series
Example. H has two 1-dimensional representations
ind : Ts 7! qcs and " : Ts 7! �1 (s 2 S).
Easy to see: kG $ ind under Irrk(G j (H ; kH))1�1 ! Irr(H).
StG = Steinberg module. Simple , [G : B]1k 6= 0. [Steinberg 1957]
soc(StG ) always simple [Tinberg 1986].
soc(Stk)$ " under Irrk(G j (H ; kH))1�1 ! Irr(H): [G. 2015]
Recall: Natural parametrisation of Irr(H) by Λ�k � IrrC(W ).
What is the label of soc(StG )$ " in Λ�k ?
(If [G : B]1k 6= 0, then Λ�
k = IrrC(W ), label is sign representation of W .)
If G = GLn(q), then W �= Sn and IrrC(W )$ fpartitions of ng.
Label of " corresponds to image of (n) under Mullineux involution.
Analogous result for classical groups G (q). Description of Mullineux
involution using crystal graph theory [Jacon arXiv:1509:03417].
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 8 / 13
One particular case: Unipotent principal series
Example. H has two 1-dimensional representations
ind : Ts 7! qcs and " : Ts 7! �1 (s 2 S).
Easy to see: kG $ ind under Irrk(G j (H ; kH))1�1 ! Irr(H).
StG = Steinberg module. Simple , [G : B]1k 6= 0. [Steinberg 1957]
soc(StG ) always simple [Tinberg 1986].
soc(Stk)$ " under Irrk(G j (H ; kH))1�1 ! Irr(H): [G. 2015]
Recall: Natural parametrisation of Irr(H) by Λ�k � IrrC(W ).
What is the label of soc(StG )$ " in Λ�k ?
(If [G : B]1k 6= 0, then Λ�
k = IrrC(W ), label is sign representation of W .)
If G = GLn(q), then W �= Sn and IrrC(W )$ fpartitions of ng.
Label of " corresponds to image of (n) under Mullineux involution.
Analogous result for classical groups G (q). Description of Mullineux
involution using crystal graph theory [Jacon arXiv:1509:03417].
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 8 / 13
One particular case: Unipotent principal series
Example. H has two 1-dimensional representations
ind : Ts 7! qcs and " : Ts 7! �1 (s 2 S).
Easy to see: kG $ ind under Irrk(G j (H ; kH))1�1 ! Irr(H).
StG = Steinberg module. Simple , [G : B]1k 6= 0. [Steinberg 1957]
soc(StG ) always simple [Tinberg 1986].
soc(Stk)$ " under Irrk(G j (H ; kH))1�1 ! Irr(H): [G. 2015]
Recall: Natural parametrisation of Irr(H) by Λ�k � IrrC(W ).
What is the label of soc(StG )$ " in Λ�k ?
(If [G : B]1k 6= 0, then Λ�
k = IrrC(W ), label is sign representation of W .)
If G = GLn(q), then W �= Sn and IrrC(W )$ fpartitions of ng.
Label of " corresponds to image of (n) under Mullineux involution.
Analogous result for classical groups G (q). Description of Mullineux
involution using crystal graph theory [Jacon arXiv:1509:03417].
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 8 / 13
One particular case: Unipotent principal series
Example. H has two 1-dimensional representations
ind : Ts 7! qcs and " : Ts 7! �1 (s 2 S).
Easy to see: kG $ ind under Irrk(G j (H ; kH))1�1 ! Irr(H).
StG = Steinberg module. Simple , [G : B]1k 6= 0. [Steinberg 1957]
soc(StG ) always simple [Tinberg 1986].
soc(Stk)$ " under Irrk(G j (H ; kH))1�1 ! Irr(H): [G. 2015]
Recall: Natural parametrisation of Irr(H) by Λ�k � IrrC(W ).
What is the label of soc(StG )$ " in Λ�k ?
(If [G : B]1k 6= 0, then Λ�
k = IrrC(W ), label is sign representation of W .)
If G = GLn(q), then W �= Sn and IrrC(W )$ fpartitions of ng.
Label of " corresponds to image of (n) under Mullineux involution.
Analogous result for classical groups G (q).
Description of Mullineux
involution using crystal graph theory [Jacon arXiv:1509:03417].
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 8 / 13
One particular case: Unipotent principal series
Example. H has two 1-dimensional representations
ind : Ts 7! qcs and " : Ts 7! �1 (s 2 S).
Easy to see: kG $ ind under Irrk(G j (H ; kH))1�1 ! Irr(H).
StG = Steinberg module. Simple , [G : B]1k 6= 0. [Steinberg 1957]
soc(StG ) always simple [Tinberg 1986].
soc(Stk)$ " under Irrk(G j (H ; kH))1�1 ! Irr(H): [G. 2015]
Recall: Natural parametrisation of Irr(H) by Λ�k � IrrC(W ).
What is the label of soc(StG )$ " in Λ�k ?
(If [G : B]1k 6= 0, then Λ�
k = IrrC(W ), label is sign representation of W .)
If G = GLn(q), then W �= Sn and IrrC(W )$ fpartitions of ng.
Label of " corresponds to image of (n) under Mullineux involution.
Analogous result for classical groups G (q). Description of Mullineux
involution using crystal graph theory [Jacon arXiv:1509:03417].
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 8 / 13
Back to general theory: Decomposition numbers
Fix `-modular system K � O � k for G = G (q),
where char(K ) = 0,
char(k) = ` > 0 and ` - q. Decomposition matrix D = (dV ;M) where
dV ;M =multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).
General idea: IrrK (G ) “known”, get hold of Irrk(G ) via D.
Deligne–Lusztig theory:
V 2 IrrK (G ) # (s) semisimple conjugacy class in G �(q).
�dV ;M
�=
D1
D2
D3
Dr
� � �
�
0
0Each Di is square and det(Di ) = �1
Rows of Di $ Ei � IrrK (G ) where
Ei # (si ) with si fixed of order prime to `
[Broue–Michel 1989 + G.–Hiss 1991]
Requires: ` “good” prime for G and
connected center or ` - jZ (G (q))j
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 9 / 13
Back to general theory: Decomposition numbers
Fix `-modular system K � O � k for G = G (q), where char(K ) = 0,
char(k) = ` > 0 and ` - q.
Decomposition matrix D = (dV ;M) where
dV ;M =multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).
General idea: IrrK (G ) “known”, get hold of Irrk(G ) via D.
Deligne–Lusztig theory:
V 2 IrrK (G ) # (s) semisimple conjugacy class in G �(q).
�dV ;M
�=
D1
D2
D3
Dr
� � �
�
0
0Each Di is square and det(Di ) = �1
Rows of Di $ Ei � IrrK (G ) where
Ei # (si ) with si fixed of order prime to `
[Broue–Michel 1989 + G.–Hiss 1991]
Requires: ` “good” prime for G and
connected center or ` - jZ (G (q))j
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 9 / 13
Back to general theory: Decomposition numbers
Fix `-modular system K � O � k for G = G (q), where char(K ) = 0,
char(k) = ` > 0 and ` - q. Decomposition matrix D = (dV ;M) where
dV ;M =
multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).
General idea: IrrK (G ) “known”, get hold of Irrk(G ) via D.
Deligne–Lusztig theory:
V 2 IrrK (G ) # (s) semisimple conjugacy class in G �(q).
�dV ;M
�=
D1
D2
D3
Dr
� � �
�
0
0Each Di is square and det(Di ) = �1
Rows of Di $ Ei � IrrK (G ) where
Ei # (si ) with si fixed of order prime to `
[Broue–Michel 1989 + G.–Hiss 1991]
Requires: ` “good” prime for G and
connected center or ` - jZ (G (q))j
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 9 / 13
Back to general theory: Decomposition numbers
Fix `-modular system K � O � k for G = G (q), where char(K ) = 0,
char(k) = ` > 0 and ` - q. Decomposition matrix D = (dV ;M) where
dV ;M =multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).
General idea: IrrK (G ) “known”, get hold of Irrk(G ) via D.
Deligne–Lusztig theory:
V 2 IrrK (G ) # (s) semisimple conjugacy class in G �(q).
�dV ;M
�=
D1
D2
D3
Dr
� � �
�
0
0Each Di is square and det(Di ) = �1
Rows of Di $ Ei � IrrK (G ) where
Ei # (si ) with si fixed of order prime to `
[Broue–Michel 1989 + G.–Hiss 1991]
Requires: ` “good” prime for G and
connected center or ` - jZ (G (q))j
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 9 / 13
Back to general theory: Decomposition numbers
Fix `-modular system K � O � k for G = G (q), where char(K ) = 0,
char(k) = ` > 0 and ` - q. Decomposition matrix D = (dV ;M) where
dV ;M =multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).
General idea:
IrrK (G ) “known”, get hold of Irrk(G ) via D.
Deligne–Lusztig theory:
V 2 IrrK (G ) # (s) semisimple conjugacy class in G �(q).
�dV ;M
�=
D1
D2
D3
Dr
� � �
�
0
0Each Di is square and det(Di ) = �1
Rows of Di $ Ei � IrrK (G ) where
Ei # (si ) with si fixed of order prime to `
[Broue–Michel 1989 + G.–Hiss 1991]
Requires: ` “good” prime for G and
connected center or ` - jZ (G (q))j
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 9 / 13
Back to general theory: Decomposition numbers
Fix `-modular system K � O � k for G = G (q), where char(K ) = 0,
char(k) = ` > 0 and ` - q. Decomposition matrix D = (dV ;M) where
dV ;M =multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).
General idea: IrrK (G ) “known”,
get hold of Irrk(G ) via D.
Deligne–Lusztig theory:
V 2 IrrK (G ) # (s) semisimple conjugacy class in G �(q).
�dV ;M
�=
D1
D2
D3
Dr
� � �
�
0
0Each Di is square and det(Di ) = �1
Rows of Di $ Ei � IrrK (G ) where
Ei # (si ) with si fixed of order prime to `
[Broue–Michel 1989 + G.–Hiss 1991]
Requires: ` “good” prime for G and
connected center or ` - jZ (G (q))j
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 9 / 13
Back to general theory: Decomposition numbers
Fix `-modular system K � O � k for G = G (q), where char(K ) = 0,
char(k) = ` > 0 and ` - q. Decomposition matrix D = (dV ;M) where
dV ;M =multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).
General idea: IrrK (G ) “known”, get hold of Irrk(G ) via D.
Deligne–Lusztig theory:
V 2 IrrK (G ) # (s) semisimple conjugacy class in G �(q).
�dV ;M
�=
D1
D2
D3
Dr
� � �
�
0
0Each Di is square and det(Di ) = �1
Rows of Di $ Ei � IrrK (G ) where
Ei # (si ) with si fixed of order prime to `
[Broue–Michel 1989 + G.–Hiss 1991]
Requires: ` “good” prime for G and
connected center or ` - jZ (G (q))j
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 9 / 13
Back to general theory: Decomposition numbers
Fix `-modular system K � O � k for G = G (q), where char(K ) = 0,
char(k) = ` > 0 and ` - q. Decomposition matrix D = (dV ;M) where
dV ;M =multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).
General idea: IrrK (G ) “known”, get hold of Irrk(G ) via D.
Deligne–Lusztig theory:
V 2 IrrK (G ) # (s) semisimple conjugacy class in G �(q).
�dV ;M
�=
D1
D2
D3
Dr
� � �
�
0
0Each Di is square and det(Di ) = �1
Rows of Di $ Ei � IrrK (G ) where
Ei # (si ) with si fixed of order prime to `
[Broue–Michel 1989 + G.–Hiss 1991]
Requires: ` “good” prime for G and
connected center or ` - jZ (G (q))j
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 9 / 13
Back to general theory: Decomposition numbers
Fix `-modular system K � O � k for G = G (q), where char(K ) = 0,
char(k) = ` > 0 and ` - q. Decomposition matrix D = (dV ;M) where
dV ;M =multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).
General idea: IrrK (G ) “known”, get hold of Irrk(G ) via D.
Deligne–Lusztig theory:
V 2 IrrK (G ) # (s) semisimple conjugacy class in G �(q).
�dV ;M
�=
D1
D2
D3
Dr
� � �
�
0
0Each Di is square and det(Di ) = �1
Rows of Di $ Ei � IrrK (G ) where
Ei # (si ) with si fixed of order prime to `
[Broue–Michel 1989 + G.–Hiss 1991]
Requires: ` “good” prime for G and
connected center or ` - jZ (G (q))j
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 9 / 13
Back to general theory: Decomposition numbers
Fix `-modular system K � O � k for G = G (q), where char(K ) = 0,
char(k) = ` > 0 and ` - q. Decomposition matrix D = (dV ;M) where
dV ;M =multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).
General idea: IrrK (G ) “known”, get hold of Irrk(G ) via D.
Deligne–Lusztig theory:
V 2 IrrK (G ) # (s) semisimple conjugacy class in G �(q).
�dV ;M
�=
D1
D2
D3
Dr
� � �
�
0
0Each Di is square and det(Di ) = �1
Rows of Di $ Ei � IrrK (G ) where
Ei # (si ) with si fixed of order prime to `
[Broue–Michel 1989 + G.–Hiss 1991]
Requires: ` “good” prime for G and
connected center or ` - jZ (G (q))j
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 9 / 13
Back to general theory: Decomposition numbers
Fix `-modular system K � O � k for G = G (q), where char(K ) = 0,
char(k) = ` > 0 and ` - q. Decomposition matrix D = (dV ;M) where
dV ;M =multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).
General idea: IrrK (G ) “known”, get hold of Irrk(G ) via D.
Deligne–Lusztig theory:
V 2 IrrK (G ) # (s) semisimple conjugacy class in G �(q).
�dV ;M
�=
D1
D2
D3
Dr
� � �
�
0
0
Each Di is square and det(Di ) = �1
Rows of Di $ Ei � IrrK (G ) where
Ei # (si ) with si fixed of order prime to `
[Broue–Michel 1989 + G.–Hiss 1991]
Requires: ` “good” prime for G and
connected center or ` - jZ (G (q))j
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 9 / 13
Back to general theory: Decomposition numbers
Fix `-modular system K � O � k for G = G (q), where char(K ) = 0,
char(k) = ` > 0 and ` - q. Decomposition matrix D = (dV ;M) where
dV ;M =multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).
General idea: IrrK (G ) “known”, get hold of Irrk(G ) via D.
Deligne–Lusztig theory:
V 2 IrrK (G ) # (s) semisimple conjugacy class in G �(q).
�dV ;M
�=
D1
D2
D3
Dr
� � �
�
0
0Each Di is square and det(Di ) = �1
Rows of Di $ Ei � IrrK (G ) where
Ei # (si ) with si fixed of order prime to `
[Broue–Michel 1989 + G.–Hiss 1991]
Requires: ` “good” prime for G and
connected center or ` - jZ (G (q))j
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 9 / 13
Back to general theory: Decomposition numbers
Fix `-modular system K � O � k for G = G (q), where char(K ) = 0,
char(k) = ` > 0 and ` - q. Decomposition matrix D = (dV ;M) where
dV ;M =multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).
General idea: IrrK (G ) “known”, get hold of Irrk(G ) via D.
Deligne–Lusztig theory:
V 2 IrrK (G ) # (s) semisimple conjugacy class in G �(q).
�dV ;M
�=
D1
D2
D3
Dr
� � �
�
0
0Each Di is square and det(Di ) = �1
Rows of Di $ Ei � IrrK (G ) where
Ei # (si ) with si fixed of order prime to `
[Broue–Michel 1989 + G.–Hiss 1991]
Requires: ` “good” prime for G and
connected center or ` - jZ (G (q))j
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 9 / 13
Back to general theory: Decomposition numbers
Fix `-modular system K � O � k for G = G (q), where char(K ) = 0,
char(k) = ` > 0 and ` - q. Decomposition matrix D = (dV ;M) where
dV ;M =multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).
General idea: IrrK (G ) “known”, get hold of Irrk(G ) via D.
Deligne–Lusztig theory:
V 2 IrrK (G ) # (s) semisimple conjugacy class in G �(q).
�dV ;M
�=
D1
D2
D3
Dr
� � �
�
0
0Each Di is square and det(Di ) = �1
Rows of Di $ Ei � IrrK (G ) where
Ei # (si ) with si fixed of order prime to `
[Broue–Michel 1989 + G.–Hiss 1991]
Requires: ` “good” prime for G and
connected center or ` - jZ (G (q))j
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 9 / 13
Back to general theory: Decomposition numbers
Fix `-modular system K � O � k for G = G (q), where char(K ) = 0,
char(k) = ` > 0 and ` - q. Decomposition matrix D = (dV ;M) where
dV ;M =multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).
General idea: IrrK (G ) “known”, get hold of Irrk(G ) via D.
Deligne–Lusztig theory:
V 2 IrrK (G ) # (s) semisimple conjugacy class in G �(q).
�dV ;M
�=
D1
D2
D3
Dr
� � �
�
0
0Each Di is square and det(Di ) = �1
Rows of Di $ Ei � IrrK (G ) where
Ei # (si ) with si fixed of order prime to `
[Broue–Michel 1989 + G.–Hiss 1991]
Requires: ` “good” prime for G and
connected center or ` - jZ (G (q))j
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 9 / 13
Back to general theory: Decomposition numbers
Fix `-modular system K � O � k for G = G (q), where char(K ) = 0,
char(k) = ` > 0 and ` - q. Decomposition matrix D = (dV ;M) where
dV ;M =multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).
General idea: IrrK (G ) “known”, get hold of Irrk(G ) via D.
Deligne–Lusztig theory:
V 2 IrrK (G ) # (s) semisimple conjugacy class in G �(q).
�dV ;M
�=
D1
D2
D3
Dr
� � �
�
0
0Each Di is square and det(Di ) = �1
Rows of Di $ Ei � IrrK (G ) where
Ei # (si ) with si fixed of order prime to `
[Broue–Michel 1989 + G.–Hiss 1991]
Requires: ` “good” prime for G and
connected center or ` - jZ (G (q))j
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 9 / 13
Back to general theory: Decomposition numbers
Focus on E1 # (s1) and D1 with s1 = 1 “unipotent KG -modules”.
M 2 Irrk(G ) is called unipotent if dV ;M 6= 0 for some V 2 E1.
D1 square matrix with rows/columns labelled by unipotent modules.
Now G = G (q) general linear, unitary, symplectic or orthogonal.
Lusztig 1977: E1 parametrised by combinatorial objects.
GLn(q) or GUn(q) : Partitions of n.
Sp2n(q) or SO2n+1(q) : Symbols of rank n and odd defect.
SO�2n(q) : Certain symbols of rank n and even defect.
Partition of E1 into HC-series determined combinatorially.
(Example: At most one cuspidal unipotent KG -module, depends only on n.)
D1 square matrix ) unipotent kG -modules in bijection with E1,
hence there should also be parametrisation by combinatorial objects.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 10 / 13
Back to general theory: Decomposition numbers
Focus on E1 # (s1) and D1 with s1 = 1 “unipotent KG -modules”.
M 2 Irrk(G ) is called unipotent if dV ;M 6= 0 for some V 2 E1.
D1 square matrix with rows/columns labelled by unipotent modules.
Now G = G (q) general linear, unitary, symplectic or orthogonal.
Lusztig 1977: E1 parametrised by combinatorial objects.
GLn(q) or GUn(q) : Partitions of n.
Sp2n(q) or SO2n+1(q) : Symbols of rank n and odd defect.
SO�2n(q) : Certain symbols of rank n and even defect.
Partition of E1 into HC-series determined combinatorially.
(Example: At most one cuspidal unipotent KG -module, depends only on n.)
D1 square matrix ) unipotent kG -modules in bijection with E1,
hence there should also be parametrisation by combinatorial objects.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 10 / 13
Back to general theory: Decomposition numbers
Focus on E1 # (s1) and D1 with s1 = 1 “unipotent KG -modules”.
M 2 Irrk(G ) is called unipotent if dV ;M 6= 0 for some V 2 E1.
D1 square matrix with rows/columns labelled by unipotent modules.
Now G = G (q) general linear, unitary, symplectic or orthogonal.
Lusztig 1977: E1 parametrised by combinatorial objects.
GLn(q) or GUn(q) : Partitions of n.
Sp2n(q) or SO2n+1(q) : Symbols of rank n and odd defect.
SO�2n(q) : Certain symbols of rank n and even defect.
Partition of E1 into HC-series determined combinatorially.
(Example: At most one cuspidal unipotent KG -module, depends only on n.)
D1 square matrix ) unipotent kG -modules in bijection with E1,
hence there should also be parametrisation by combinatorial objects.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 10 / 13
Back to general theory: Decomposition numbers
Focus on E1 # (s1) and D1 with s1 = 1 “unipotent KG -modules”.
M 2 Irrk(G ) is called unipotent if dV ;M 6= 0 for some V 2 E1.
D1 square matrix with rows/columns labelled by unipotent modules.
Now G = G (q) general linear, unitary, symplectic or orthogonal.
Lusztig 1977: E1 parametrised by combinatorial objects.
GLn(q) or GUn(q) : Partitions of n.
Sp2n(q) or SO2n+1(q) : Symbols of rank n and odd defect.
SO�2n(q) : Certain symbols of rank n and even defect.
Partition of E1 into HC-series determined combinatorially.
(Example: At most one cuspidal unipotent KG -module, depends only on n.)
D1 square matrix ) unipotent kG -modules in bijection with E1,
hence there should also be parametrisation by combinatorial objects.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 10 / 13
Back to general theory: Decomposition numbers
Focus on E1 # (s1) and D1 with s1 = 1 “unipotent KG -modules”.
M 2 Irrk(G ) is called unipotent if dV ;M 6= 0 for some V 2 E1.
D1 square matrix with rows/columns labelled by unipotent modules.
Now G = G (q) general linear, unitary, symplectic or orthogonal.
Lusztig 1977: E1 parametrised by combinatorial objects.
GLn(q) or GUn(q) : Partitions of n.
Sp2n(q) or SO2n+1(q) : Symbols of rank n and odd defect.
SO�2n(q) : Certain symbols of rank n and even defect.
Partition of E1 into HC-series determined combinatorially.
(Example: At most one cuspidal unipotent KG -module, depends only on n.)
D1 square matrix ) unipotent kG -modules in bijection with E1,
hence there should also be parametrisation by combinatorial objects.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 10 / 13
Back to general theory: Decomposition numbers
Focus on E1 # (s1) and D1 with s1 = 1 “unipotent KG -modules”.
M 2 Irrk(G ) is called unipotent if dV ;M 6= 0 for some V 2 E1.
D1 square matrix with rows/columns labelled by unipotent modules.
Now G = G (q) general linear, unitary, symplectic or orthogonal.
Lusztig 1977: E1 parametrised by combinatorial objects.
GLn(q) or GUn(q) : Partitions of n.
Sp2n(q) or SO2n+1(q) : Symbols of rank n and odd defect.
SO�2n(q) : Certain symbols of rank n and even defect.
Partition of E1 into HC-series determined combinatorially.
(Example: At most one cuspidal unipotent KG -module, depends only on n.)
D1 square matrix ) unipotent kG -modules in bijection with E1,
hence there should also be parametrisation by combinatorial objects.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 10 / 13
Back to general theory: Decomposition numbers
Focus on E1 # (s1) and D1 with s1 = 1 “unipotent KG -modules”.
M 2 Irrk(G ) is called unipotent if dV ;M 6= 0 for some V 2 E1.
D1 square matrix with rows/columns labelled by unipotent modules.
Now G = G (q) general linear, unitary, symplectic or orthogonal.
Lusztig 1977: E1 parametrised by combinatorial objects.
GLn(q) or GUn(q) : Partitions of n.
Sp2n(q) or SO2n+1(q) : Symbols of rank n and odd defect.
SO�2n(q) : Certain symbols of rank n and even defect.
Partition of E1 into HC-series determined combinatorially.
(Example: At most one cuspidal unipotent KG -module, depends only on n.)
D1 square matrix ) unipotent kG -modules in bijection with E1,
hence there should also be parametrisation by combinatorial objects.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 10 / 13
Back to general theory: Decomposition numbers
Focus on E1 # (s1) and D1 with s1 = 1 “unipotent KG -modules”.
M 2 Irrk(G ) is called unipotent if dV ;M 6= 0 for some V 2 E1.
D1 square matrix with rows/columns labelled by unipotent modules.
Now G = G (q) general linear, unitary, symplectic or orthogonal.
Lusztig 1977: E1 parametrised by combinatorial objects.
GLn(q) or GUn(q) : Partitions of n.
Sp2n(q) or SO2n+1(q) : Symbols of rank n and odd defect.
SO�2n(q) : Certain symbols of rank n and even defect.
Partition of E1 into HC-series determined combinatorially.
(Example: At most one cuspidal unipotent KG -module, depends only on n.)
D1 square matrix ) unipotent kG -modules in bijection with E1,
hence there should also be parametrisation by combinatorial objects.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 10 / 13
Back to general theory: Decomposition numbers
Focus on E1 # (s1) and D1 with s1 = 1 “unipotent KG -modules”.
M 2 Irrk(G ) is called unipotent if dV ;M 6= 0 for some V 2 E1.
D1 square matrix with rows/columns labelled by unipotent modules.
Now G = G (q) general linear, unitary, symplectic or orthogonal.
Lusztig 1977: E1 parametrised by combinatorial objects.
GLn(q) or GUn(q) : Partitions of n.
Sp2n(q) or SO2n+1(q) : Symbols of rank n and odd defect.
SO�2n(q) : Certain symbols of rank n and even defect.
Partition of E1 into HC-series determined combinatorially.
(Example: At most one cuspidal unipotent KG -module, depends only on n.)
D1 square matrix ) unipotent kG -modules in bijection with E1,
hence there should also be parametrisation by combinatorial objects.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 10 / 13
Back to general theory: Decomposition numbers
Focus on E1 # (s1) and D1 with s1 = 1 “unipotent KG -modules”.
M 2 Irrk(G ) is called unipotent if dV ;M 6= 0 for some V 2 E1.
D1 square matrix with rows/columns labelled by unipotent modules.
Now G = G (q) general linear, unitary, symplectic or orthogonal.
Lusztig 1977: E1 parametrised by combinatorial objects.
GLn(q) or GUn(q) : Partitions of n.
Sp2n(q) or SO2n+1(q) : Symbols of rank n and odd defect.
SO�2n(q) : Certain symbols of rank n and even defect.
Partition of E1 into HC-series determined combinatorially.
(Example: At most one cuspidal unipotent KG -module, depends only on n.)
D1 square matrix ) unipotent kG -modules in bijection with E1,
hence there should also be parametrisation by combinatorial objects.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 10 / 13
Back to general theory: Decomposition numbers
Focus on E1 # (s1) and D1 with s1 = 1 “unipotent KG -modules”.
M 2 Irrk(G ) is called unipotent if dV ;M 6= 0 for some V 2 E1.
D1 square matrix with rows/columns labelled by unipotent modules.
Now G = G (q) general linear, unitary, symplectic or orthogonal.
Lusztig 1977: E1 parametrised by combinatorial objects.
GLn(q) or GUn(q) : Partitions of n.
Sp2n(q) or SO2n+1(q) : Symbols of rank n and odd defect.
SO�2n(q) : Certain symbols of rank n and even defect.
Partition of E1 into HC-series determined combinatorially.
(Example: At most one cuspidal unipotent KG -module, depends only on n.)
D1 square matrix
) unipotent kG -modules in bijection with E1,
hence there should also be parametrisation by combinatorial objects.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 10 / 13
Back to general theory: Decomposition numbers
Focus on E1 # (s1) and D1 with s1 = 1 “unipotent KG -modules”.
M 2 Irrk(G ) is called unipotent if dV ;M 6= 0 for some V 2 E1.
D1 square matrix with rows/columns labelled by unipotent modules.
Now G = G (q) general linear, unitary, symplectic or orthogonal.
Lusztig 1977: E1 parametrised by combinatorial objects.
GLn(q) or GUn(q) : Partitions of n.
Sp2n(q) or SO2n+1(q) : Symbols of rank n and odd defect.
SO�2n(q) : Certain symbols of rank n and even defect.
Partition of E1 into HC-series determined combinatorially.
(Example: At most one cuspidal unipotent KG -module, depends only on n.)
D1 square matrix ) unipotent kG -modules in bijection with E1,
hence there should also be parametrisation by combinatorial objects.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 10 / 13
Back to general theory: Decomposition numbers
Focus on E1 # (s1) and D1 with s1 = 1 “unipotent KG -modules”.
M 2 Irrk(G ) is called unipotent if dV ;M 6= 0 for some V 2 E1.
D1 square matrix with rows/columns labelled by unipotent modules.
Now G = G (q) general linear, unitary, symplectic or orthogonal.
Lusztig 1977: E1 parametrised by combinatorial objects.
GLn(q) or GUn(q) : Partitions of n.
Sp2n(q) or SO2n+1(q) : Symbols of rank n and odd defect.
SO�2n(q) : Certain symbols of rank n and even defect.
Partition of E1 into HC-series determined combinatorially.
(Example: At most one cuspidal unipotent KG -module, depends only on n.)
D1 square matrix ) unipotent kG -modules in bijection with E1,
hence there should also be parametrisation by combinatorial objects.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 10 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV , and order E1
according to increasing a-invariant. Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s] and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q). Then
E1 = fV � j � ` ng, ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series? When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV , and order E1
according to increasing a-invariant. Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s] and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q). Then
E1 = fV � j � ` ng, ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series? When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV ,
and order E1
according to increasing a-invariant. Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s] and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q). Then
E1 = fV � j � ` ng, ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series? When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV , and order E1
according to increasing a-invariant.
Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s] and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q). Then
E1 = fV � j � ` ng, ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series? When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV , and order E1
according to increasing a-invariant. Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s] and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q). Then
E1 = fV � j � ` ng, ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series? When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV , and order E1
according to increasing a-invariant. Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s] and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q). Then
E1 = fV � j � ` ng, ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series? When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV , and order E1
according to increasing a-invariant. Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s]
and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q). Then
E1 = fV � j � ` ng, ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series? When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV , and order E1
according to increasing a-invariant. Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s] and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q). Then
E1 = fV � j � ` ng, ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series? When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV , and order E1
according to increasing a-invariant. Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s] and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q).
Then
E1 = fV � j � ` ng, ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series? When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV , and order E1
according to increasing a-invariant. Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s] and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q). Then
E1 = fV � j � ` ng,
ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series? When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV , and order E1
according to increasing a-invariant. Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s] and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q). Then
E1 = fV � j � ` ng, ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series? When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV , and order E1
according to increasing a-invariant. Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s] and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q). Then
E1 = fV � j � ` ng, ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series? When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV , and order E1
according to increasing a-invariant. Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s] and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q). Then
E1 = fV � j � ` ng, ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n
with d�� = 1 and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series? When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV , and order E1
according to increasing a-invariant. Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s] and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q). Then
E1 = fV � j � ` ng, ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n with d�� = 1
and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series? When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV , and order E1
according to increasing a-invariant. Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s] and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q). Then
E1 = fV � j � ` ng, ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series? When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV , and order E1
according to increasing a-invariant. Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s] and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q). Then
E1 = fV � j � ` ng, ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series?
When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
Back to general theory: Decomposition numbers
Conjecture. [G. 1990, G.–Hiss 1997]
G = G (q) finite group of Lie type, ` “good” + condition on center.
Consider Lusztig’s a-function E1 ! N0, V 7! aV , and order E1
according to increasing a-invariant. Then there is a unique bijection
E11�1 ! unipotent kG -modules
such that D1 is lower triangular with 1 on the diagonal.
Proved for GLn(q) [Dipper–James 1980/90s] and GUn(q) [G. 1990].
Assume now G (q) = GLn(q) or GUn(q). Then
E1 = fV � j � ` ng, ordered by reverse dominance order.
Unipotent kG -modules fM� j � ` ng.
D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.
Partition of fM� j � ` ng into HC-series? When is M� cuspidal?
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 11 / 13
The Gerber–Hiss–Jacon conjecture
Problem solved for GLn(q) [Dipper–James 1980/90s, Dipper–Du 1997]
Fix � 2 f0; 1g and define Gm := GU2m+�(q) for m = 0; 1; 2; : : :.
Usual Mackey system LGm = f(L;P) j P parabolic, L = P=UPg.Here, L �= Gm0 � direct product of GLni (q
2) for various ni ; m = m0 +P
i ni .
Gerber–Hiss–Jacon 2015: “Weak” Mackey systemI L0Gm
= f(L;P) 2 LGm j L = P=UP “pure”, i.e., all ni = 1g:
I Obtain weak HC-series, weakly cuspidal modules.
The HC-branching graph Gq;` [Gerber–Hiss–Jacon 2015]
Vertices: all partitions � ` 2m + � for all m > 0.
Directed edge �! � if:
I j�j = 2m + � and j�j = j�j + 2 for some m > 0;
I M� factor module of RGm+1
Gm�GL1(q2)(M�).
Then: Sources $ weakly cuspidal modules; paths $ weak HC-series.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 12 / 13
The Gerber–Hiss–Jacon conjecture
Problem solved for GLn(q) [Dipper–James 1980/90s, Dipper–Du 1997]
Fix � 2 f0; 1g and define Gm := GU2m+�(q) for m = 0; 1; 2; : : :.
Usual Mackey system LGm = f(L;P) j P parabolic, L = P=UPg.Here, L �= Gm0 � direct product of GLni (q
2) for various ni ; m = m0 +P
i ni .
Gerber–Hiss–Jacon 2015: “Weak” Mackey systemI L0Gm
= f(L;P) 2 LGm j L = P=UP “pure”, i.e., all ni = 1g:
I Obtain weak HC-series, weakly cuspidal modules.
The HC-branching graph Gq;` [Gerber–Hiss–Jacon 2015]
Vertices: all partitions � ` 2m + � for all m > 0.
Directed edge �! � if:
I j�j = 2m + � and j�j = j�j + 2 for some m > 0;
I M� factor module of RGm+1
Gm�GL1(q2)(M�).
Then: Sources $ weakly cuspidal modules; paths $ weak HC-series.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 12 / 13
The Gerber–Hiss–Jacon conjecture
Problem solved for GLn(q) [Dipper–James 1980/90s, Dipper–Du 1997]
Fix � 2 f0; 1g and define Gm := GU2m+�(q) for m = 0; 1; 2; : : :.
Usual Mackey system LGm = f(L;P) j P parabolic, L = P=UPg.
Here, L �= Gm0 � direct product of GLni (q2) for various ni ; m = m0 +
Pi ni .
Gerber–Hiss–Jacon 2015: “Weak” Mackey systemI L0Gm
= f(L;P) 2 LGm j L = P=UP “pure”, i.e., all ni = 1g:
I Obtain weak HC-series, weakly cuspidal modules.
The HC-branching graph Gq;` [Gerber–Hiss–Jacon 2015]
Vertices: all partitions � ` 2m + � for all m > 0.
Directed edge �! � if:
I j�j = 2m + � and j�j = j�j + 2 for some m > 0;
I M� factor module of RGm+1
Gm�GL1(q2)(M�).
Then: Sources $ weakly cuspidal modules; paths $ weak HC-series.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 12 / 13
The Gerber–Hiss–Jacon conjecture
Problem solved for GLn(q) [Dipper–James 1980/90s, Dipper–Du 1997]
Fix � 2 f0; 1g and define Gm := GU2m+�(q) for m = 0; 1; 2; : : :.
Usual Mackey system LGm = f(L;P) j P parabolic, L = P=UPg.Here, L �= Gm0 � direct product of GLni (q
2) for various ni ; m = m0 +P
i ni .
Gerber–Hiss–Jacon 2015: “Weak” Mackey systemI L0Gm
= f(L;P) 2 LGm j L = P=UP “pure”, i.e., all ni = 1g:
I Obtain weak HC-series, weakly cuspidal modules.
The HC-branching graph Gq;` [Gerber–Hiss–Jacon 2015]
Vertices: all partitions � ` 2m + � for all m > 0.
Directed edge �! � if:
I j�j = 2m + � and j�j = j�j + 2 for some m > 0;
I M� factor module of RGm+1
Gm�GL1(q2)(M�).
Then: Sources $ weakly cuspidal modules; paths $ weak HC-series.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 12 / 13
The Gerber–Hiss–Jacon conjecture
Problem solved for GLn(q) [Dipper–James 1980/90s, Dipper–Du 1997]
Fix � 2 f0; 1g and define Gm := GU2m+�(q) for m = 0; 1; 2; : : :.
Usual Mackey system LGm = f(L;P) j P parabolic, L = P=UPg.Here, L �= Gm0 � direct product of GLni (q
2) for various ni ; m = m0 +P
i ni .
Gerber–Hiss–Jacon 2015: “Weak” Mackey system
I L0Gm= f(L;P) 2 LGm j L = P=UP “pure”, i.e., all ni = 1g:
I Obtain weak HC-series, weakly cuspidal modules.
The HC-branching graph Gq;` [Gerber–Hiss–Jacon 2015]
Vertices: all partitions � ` 2m + � for all m > 0.
Directed edge �! � if:
I j�j = 2m + � and j�j = j�j + 2 for some m > 0;
I M� factor module of RGm+1
Gm�GL1(q2)(M�).
Then: Sources $ weakly cuspidal modules; paths $ weak HC-series.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 12 / 13
The Gerber–Hiss–Jacon conjecture
Problem solved for GLn(q) [Dipper–James 1980/90s, Dipper–Du 1997]
Fix � 2 f0; 1g and define Gm := GU2m+�(q) for m = 0; 1; 2; : : :.
Usual Mackey system LGm = f(L;P) j P parabolic, L = P=UPg.Here, L �= Gm0 � direct product of GLni (q
2) for various ni ; m = m0 +P
i ni .
Gerber–Hiss–Jacon 2015: “Weak” Mackey systemI L0Gm
= f(L;P) 2 LGm j L = P=UP “pure”, i.e., all ni = 1g:
I Obtain weak HC-series, weakly cuspidal modules.
The HC-branching graph Gq;` [Gerber–Hiss–Jacon 2015]
Vertices: all partitions � ` 2m + � for all m > 0.
Directed edge �! � if:
I j�j = 2m + � and j�j = j�j + 2 for some m > 0;
I M� factor module of RGm+1
Gm�GL1(q2)(M�).
Then: Sources $ weakly cuspidal modules; paths $ weak HC-series.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 12 / 13
The Gerber–Hiss–Jacon conjecture
Problem solved for GLn(q) [Dipper–James 1980/90s, Dipper–Du 1997]
Fix � 2 f0; 1g and define Gm := GU2m+�(q) for m = 0; 1; 2; : : :.
Usual Mackey system LGm = f(L;P) j P parabolic, L = P=UPg.Here, L �= Gm0 � direct product of GLni (q
2) for various ni ; m = m0 +P
i ni .
Gerber–Hiss–Jacon 2015: “Weak” Mackey systemI L0Gm
= f(L;P) 2 LGm j L = P=UP “pure”, i.e., all ni = 1g:
I Obtain weak HC-series, weakly cuspidal modules.
The HC-branching graph Gq;` [Gerber–Hiss–Jacon 2015]
Vertices: all partitions � ` 2m + � for all m > 0.
Directed edge �! � if:
I j�j = 2m + � and j�j = j�j + 2 for some m > 0;
I M� factor module of RGm+1
Gm�GL1(q2)(M�).
Then: Sources $ weakly cuspidal modules; paths $ weak HC-series.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 12 / 13
The Gerber–Hiss–Jacon conjecture
Problem solved for GLn(q) [Dipper–James 1980/90s, Dipper–Du 1997]
Fix � 2 f0; 1g and define Gm := GU2m+�(q) for m = 0; 1; 2; : : :.
Usual Mackey system LGm = f(L;P) j P parabolic, L = P=UPg.Here, L �= Gm0 � direct product of GLni (q
2) for various ni ; m = m0 +P
i ni .
Gerber–Hiss–Jacon 2015: “Weak” Mackey systemI L0Gm
= f(L;P) 2 LGm j L = P=UP “pure”, i.e., all ni = 1g:
I Obtain weak HC-series, weakly cuspidal modules.
The HC-branching graph Gq;` [Gerber–Hiss–Jacon 2015]
Vertices: all partitions � ` 2m + � for all m > 0.
Directed edge �! � if:
I j�j = 2m + � and j�j = j�j + 2 for some m > 0;
I M� factor module of RGm+1
Gm�GL1(q2)(M�).
Then: Sources $ weakly cuspidal modules; paths $ weak HC-series.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 12 / 13
The Gerber–Hiss–Jacon conjecture
Problem solved for GLn(q) [Dipper–James 1980/90s, Dipper–Du 1997]
Fix � 2 f0; 1g and define Gm := GU2m+�(q) for m = 0; 1; 2; : : :.
Usual Mackey system LGm = f(L;P) j P parabolic, L = P=UPg.Here, L �= Gm0 � direct product of GLni (q
2) for various ni ; m = m0 +P
i ni .
Gerber–Hiss–Jacon 2015: “Weak” Mackey systemI L0Gm
= f(L;P) 2 LGm j L = P=UP “pure”, i.e., all ni = 1g:
I Obtain weak HC-series, weakly cuspidal modules.
The HC-branching graph Gq;` [Gerber–Hiss–Jacon 2015]
Vertices: all partitions � ` 2m + � for all m > 0.
Directed edge �! � if:
I j�j = 2m + � and j�j = j�j + 2 for some m > 0;
I M� factor module of RGm+1
Gm�GL1(q2)(M�).
Then: Sources $ weakly cuspidal modules; paths $ weak HC-series.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 12 / 13
The Gerber–Hiss–Jacon conjecture
Problem solved for GLn(q) [Dipper–James 1980/90s, Dipper–Du 1997]
Fix � 2 f0; 1g and define Gm := GU2m+�(q) for m = 0; 1; 2; : : :.
Usual Mackey system LGm = f(L;P) j P parabolic, L = P=UPg.Here, L �= Gm0 � direct product of GLni (q
2) for various ni ; m = m0 +P
i ni .
Gerber–Hiss–Jacon 2015: “Weak” Mackey systemI L0Gm
= f(L;P) 2 LGm j L = P=UP “pure”, i.e., all ni = 1g:
I Obtain weak HC-series, weakly cuspidal modules.
The HC-branching graph Gq;` [Gerber–Hiss–Jacon 2015]
Vertices: all partitions � ` 2m + � for all m > 0.
Directed edge �! � if:
I j�j = 2m + � and j�j = j�j + 2 for some m > 0;
I M� factor module of RGm+1
Gm�GL1(q2)(M�).
Then: Sources $ weakly cuspidal modules; paths $ weak HC-series.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 12 / 13
The Gerber–Hiss–Jacon conjecture
Problem solved for GLn(q) [Dipper–James 1980/90s, Dipper–Du 1997]
Fix � 2 f0; 1g and define Gm := GU2m+�(q) for m = 0; 1; 2; : : :.
Usual Mackey system LGm = f(L;P) j P parabolic, L = P=UPg.Here, L �= Gm0 � direct product of GLni (q
2) for various ni ; m = m0 +P
i ni .
Gerber–Hiss–Jacon 2015: “Weak” Mackey systemI L0Gm
= f(L;P) 2 LGm j L = P=UP “pure”, i.e., all ni = 1g:
I Obtain weak HC-series, weakly cuspidal modules.
The HC-branching graph Gq;` [Gerber–Hiss–Jacon 2015]
Vertices: all partitions � ` 2m + � for all m > 0.
Directed edge �! � if:
I j�j = 2m + � and j�j = j�j + 2 for some m > 0;
I M� factor module of RGm+1
Gm�GL1(q2)(M�).
Then: Sources $ weakly cuspidal modules; paths $ weak HC-series.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 12 / 13
The Gerber–Hiss–Jacon conjecture
Problem solved for GLn(q) [Dipper–James 1980/90s, Dipper–Du 1997]
Fix � 2 f0; 1g and define Gm := GU2m+�(q) for m = 0; 1; 2; : : :.
Usual Mackey system LGm = f(L;P) j P parabolic, L = P=UPg.Here, L �= Gm0 � direct product of GLni (q
2) for various ni ; m = m0 +P
i ni .
Gerber–Hiss–Jacon 2015: “Weak” Mackey systemI L0Gm
= f(L;P) 2 LGm j L = P=UP “pure”, i.e., all ni = 1g:
I Obtain weak HC-series, weakly cuspidal modules.
The HC-branching graph Gq;` [Gerber–Hiss–Jacon 2015]
Vertices: all partitions � ` 2m + � for all m > 0.
Directed edge �! � if:
I j�j = 2m + � and j�j = j�j + 2 for some m > 0;
I M� factor module of RGm+1
Gm�GL1(q2)(M�).
Then: Sources $ weakly cuspidal modules; paths $ weak HC-series.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 12 / 13
The Gerber–Hiss–Jacon conjecture
Problem solved for GLn(q) [Dipper–James 1980/90s, Dipper–Du 1997]
Fix � 2 f0; 1g and define Gm := GU2m+�(q) for m = 0; 1; 2; : : :.
Usual Mackey system LGm = f(L;P) j P parabolic, L = P=UPg.Here, L �= Gm0 � direct product of GLni (q
2) for various ni ; m = m0 +P
i ni .
Gerber–Hiss–Jacon 2015: “Weak” Mackey systemI L0Gm
= f(L;P) 2 LGm j L = P=UP “pure”, i.e., all ni = 1g:
I Obtain weak HC-series, weakly cuspidal modules.
The HC-branching graph Gq;` [Gerber–Hiss–Jacon 2015]
Vertices: all partitions � ` 2m + � for all m > 0.
Directed edge �! � if:
I j�j = 2m + � and j�j = j�j + 2 for some m > 0;
I M� factor module of RGm+1
Gm�GL1(q2)(M�).
Then: Sources $ weakly cuspidal modules;
paths $ weak HC-series.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 12 / 13
The Gerber–Hiss–Jacon conjecture
Problem solved for GLn(q) [Dipper–James 1980/90s, Dipper–Du 1997]
Fix � 2 f0; 1g and define Gm := GU2m+�(q) for m = 0; 1; 2; : : :.
Usual Mackey system LGm = f(L;P) j P parabolic, L = P=UPg.Here, L �= Gm0 � direct product of GLni (q
2) for various ni ; m = m0 +P
i ni .
Gerber–Hiss–Jacon 2015: “Weak” Mackey systemI L0Gm
= f(L;P) 2 LGm j L = P=UP “pure”, i.e., all ni = 1g:
I Obtain weak HC-series, weakly cuspidal modules.
The HC-branching graph Gq;` [Gerber–Hiss–Jacon 2015]
Vertices: all partitions � ` 2m + � for all m > 0.
Directed edge �! � if:
I j�j = 2m + � and j�j = j�j + 2 for some m > 0;
I M� factor module of RGm+1
Gm�GL1(q2)(M�).
Then: Sources $ weakly cuspidal modules; paths $ weak HC-series.
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 12 / 13
The Gerber–Hiss–Jacon conjecture
Theorem. [Dudas–Vasserot–Varagnolo arXiv:1509:03269], G.H.J. conjecture.
Let e = multiplicative order of �q mod `. Assume e odd and e > 3
() ` not “linear”). Then the HC-branching graph coincides with the
union of the crystal graphs of certain level 2 Fock spaces, each of
which corresponds to an ordinary HC-series of simple KGm-modules.
This gives explicit combinatorial description of weakly cuspidal
modules and weak HC-series of finite unitary groups Gm.
M�;M� belong to the same weak HC-series of kGm-modules )
V �;V � belong to the same ordinary HC-series of KGm-modules.
D.V.V. also sketch strategies:
I to obtain “proper” cuspidal modules and HC-series of Gm.
I to prove analogous results for symplectic and orthogonal groups
(by-passing missing triangularity conjecture in these cases).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 13 / 13
The Gerber–Hiss–Jacon conjecture
Theorem. [Dudas–Vasserot–Varagnolo arXiv:1509:03269], G.H.J. conjecture.
Let e = multiplicative order of �q mod `. Assume e odd and e > 3
() ` not “linear”).
Then the HC-branching graph coincides with the
union of the crystal graphs of certain level 2 Fock spaces, each of
which corresponds to an ordinary HC-series of simple KGm-modules.
This gives explicit combinatorial description of weakly cuspidal
modules and weak HC-series of finite unitary groups Gm.
M�;M� belong to the same weak HC-series of kGm-modules )
V �;V � belong to the same ordinary HC-series of KGm-modules.
D.V.V. also sketch strategies:
I to obtain “proper” cuspidal modules and HC-series of Gm.
I to prove analogous results for symplectic and orthogonal groups
(by-passing missing triangularity conjecture in these cases).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 13 / 13
The Gerber–Hiss–Jacon conjecture
Theorem. [Dudas–Vasserot–Varagnolo arXiv:1509:03269], G.H.J. conjecture.
Let e = multiplicative order of �q mod `. Assume e odd and e > 3
() ` not “linear”). Then the HC-branching graph coincides with the
union of the crystal graphs of certain level 2 Fock spaces,
each of
which corresponds to an ordinary HC-series of simple KGm-modules.
This gives explicit combinatorial description of weakly cuspidal
modules and weak HC-series of finite unitary groups Gm.
M�;M� belong to the same weak HC-series of kGm-modules )
V �;V � belong to the same ordinary HC-series of KGm-modules.
D.V.V. also sketch strategies:
I to obtain “proper” cuspidal modules and HC-series of Gm.
I to prove analogous results for symplectic and orthogonal groups
(by-passing missing triangularity conjecture in these cases).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 13 / 13
The Gerber–Hiss–Jacon conjecture
Theorem. [Dudas–Vasserot–Varagnolo arXiv:1509:03269], G.H.J. conjecture.
Let e = multiplicative order of �q mod `. Assume e odd and e > 3
() ` not “linear”). Then the HC-branching graph coincides with the
union of the crystal graphs of certain level 2 Fock spaces, each of
which corresponds to an ordinary HC-series of simple KGm-modules.
This gives explicit combinatorial description of weakly cuspidal
modules and weak HC-series of finite unitary groups Gm.
M�;M� belong to the same weak HC-series of kGm-modules )
V �;V � belong to the same ordinary HC-series of KGm-modules.
D.V.V. also sketch strategies:
I to obtain “proper” cuspidal modules and HC-series of Gm.
I to prove analogous results for symplectic and orthogonal groups
(by-passing missing triangularity conjecture in these cases).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 13 / 13
The Gerber–Hiss–Jacon conjecture
Theorem. [Dudas–Vasserot–Varagnolo arXiv:1509:03269], G.H.J. conjecture.
Let e = multiplicative order of �q mod `. Assume e odd and e > 3
() ` not “linear”). Then the HC-branching graph coincides with the
union of the crystal graphs of certain level 2 Fock spaces, each of
which corresponds to an ordinary HC-series of simple KGm-modules.
This gives explicit combinatorial description of weakly cuspidal
modules and weak HC-series of finite unitary groups Gm.
M�;M� belong to the same weak HC-series of kGm-modules )
V �;V � belong to the same ordinary HC-series of KGm-modules.
D.V.V. also sketch strategies:
I to obtain “proper” cuspidal modules and HC-series of Gm.
I to prove analogous results for symplectic and orthogonal groups
(by-passing missing triangularity conjecture in these cases).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 13 / 13
The Gerber–Hiss–Jacon conjecture
Theorem. [Dudas–Vasserot–Varagnolo arXiv:1509:03269], G.H.J. conjecture.
Let e = multiplicative order of �q mod `. Assume e odd and e > 3
() ` not “linear”). Then the HC-branching graph coincides with the
union of the crystal graphs of certain level 2 Fock spaces, each of
which corresponds to an ordinary HC-series of simple KGm-modules.
This gives explicit combinatorial description of weakly cuspidal
modules and weak HC-series of finite unitary groups Gm.
M�;M� belong to the same weak HC-series of kGm-modules
)
V �;V � belong to the same ordinary HC-series of KGm-modules.
D.V.V. also sketch strategies:
I to obtain “proper” cuspidal modules and HC-series of Gm.
I to prove analogous results for symplectic and orthogonal groups
(by-passing missing triangularity conjecture in these cases).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 13 / 13
The Gerber–Hiss–Jacon conjecture
Theorem. [Dudas–Vasserot–Varagnolo arXiv:1509:03269], G.H.J. conjecture.
Let e = multiplicative order of �q mod `. Assume e odd and e > 3
() ` not “linear”). Then the HC-branching graph coincides with the
union of the crystal graphs of certain level 2 Fock spaces, each of
which corresponds to an ordinary HC-series of simple KGm-modules.
This gives explicit combinatorial description of weakly cuspidal
modules and weak HC-series of finite unitary groups Gm.
M�;M� belong to the same weak HC-series of kGm-modules )
V �;V � belong to the same ordinary HC-series of KGm-modules.
D.V.V. also sketch strategies:
I to obtain “proper” cuspidal modules and HC-series of Gm.
I to prove analogous results for symplectic and orthogonal groups
(by-passing missing triangularity conjecture in these cases).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 13 / 13
The Gerber–Hiss–Jacon conjecture
Theorem. [Dudas–Vasserot–Varagnolo arXiv:1509:03269], G.H.J. conjecture.
Let e = multiplicative order of �q mod `. Assume e odd and e > 3
() ` not “linear”). Then the HC-branching graph coincides with the
union of the crystal graphs of certain level 2 Fock spaces, each of
which corresponds to an ordinary HC-series of simple KGm-modules.
This gives explicit combinatorial description of weakly cuspidal
modules and weak HC-series of finite unitary groups Gm.
M�;M� belong to the same weak HC-series of kGm-modules )
V �;V � belong to the same ordinary HC-series of KGm-modules.
D.V.V. also sketch strategies:
I to obtain “proper” cuspidal modules and HC-series of Gm.
I to prove analogous results for symplectic and orthogonal groups
(by-passing missing triangularity conjecture in these cases).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 13 / 13
The Gerber–Hiss–Jacon conjecture
Theorem. [Dudas–Vasserot–Varagnolo arXiv:1509:03269], G.H.J. conjecture.
Let e = multiplicative order of �q mod `. Assume e odd and e > 3
() ` not “linear”). Then the HC-branching graph coincides with the
union of the crystal graphs of certain level 2 Fock spaces, each of
which corresponds to an ordinary HC-series of simple KGm-modules.
This gives explicit combinatorial description of weakly cuspidal
modules and weak HC-series of finite unitary groups Gm.
M�;M� belong to the same weak HC-series of kGm-modules )
V �;V � belong to the same ordinary HC-series of KGm-modules.
D.V.V. also sketch strategies:
I to obtain “proper” cuspidal modules and HC-series of Gm.
I to prove analogous results for symplectic and orthogonal groups
(by-passing missing triangularity conjecture in these cases).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 13 / 13
The Gerber–Hiss–Jacon conjecture
Theorem. [Dudas–Vasserot–Varagnolo arXiv:1509:03269], G.H.J. conjecture.
Let e = multiplicative order of �q mod `. Assume e odd and e > 3
() ` not “linear”). Then the HC-branching graph coincides with the
union of the crystal graphs of certain level 2 Fock spaces, each of
which corresponds to an ordinary HC-series of simple KGm-modules.
This gives explicit combinatorial description of weakly cuspidal
modules and weak HC-series of finite unitary groups Gm.
M�;M� belong to the same weak HC-series of kGm-modules )
V �;V � belong to the same ordinary HC-series of KGm-modules.
D.V.V. also sketch strategies:
I to obtain “proper” cuspidal modules and HC-series of Gm.
I to prove analogous results for symplectic and orthogonal groups
(by-passing missing triangularity conjecture in these cases).
Meinolf Geck (Universitat Stuttgart) Modular Harish-Chandra series Shanghai, December 2015 13 / 13