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, : : :

The basic set-up

I Unitary GUn(q),

The basic set-up

I Unitary GUn(q),

The basic set-up

I Unitary GUn(q),

The basic set-up

I Unitary GUn(q),

Describe Irrk(G (q)):

Classification (by combinatorial objects),

The basic set-up

I Unitary GUn(q),

The basic set-up

I Unitary GUn(q),

”Defining characteristic case”: p = ` > 0

The basic set-up

I Unitary GUn(q),

”Defining characteristic case”: p = ` > 0

Algebraic group techniques (highest weight theory, : : :).

The basic set-up

I Unitary GUn(q),

This talk: ”Non-defining characteristic case” ` 6= p

The basic set-up

I Unitary GUn(q),

This talk: ”Non-defining characteristic case” ` 6= p

Harish-Chandra theory: Inductive approach.

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).

Classical case: G any finite group, L � G subgroup

Right adjointness

L;P(Y ))

Right adjointness

L;P(Y ))

Right adjointness

L;P(Y ))

Harish-Chandra induction and restriction.

Let P � G be a subgroup

Right adjointness

L;P(Y ))

Right adjointness

L;P(Y ))

Right adjointness

L;P(Y ))

[first inflate to P, then

[first restrict to P, then take

induce from P to G ]

fixed points under ker(P�L)]

Right adjointness

L;P(Y ))

Right adjointness

L;P(Y ))

Right adjointness

L;P(Y ))

Right adjointness

L;P(Y ))

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)

(M ;Q) � (L;P)def() UP � UQ � Q � P .

char(k) = ` 6= p

(M ;Q) � (L;P)def() UP � UQ � Q � P .

char(k) = ` 6= p

(M ;Q) � (L;P)

def() UP � UQ � Q � P .

char(k) = ` 6= p

(M ;Q) � (L;P)def() UP � UQ � Q � P .

char(k) = ` 6= p

(M ;Q) � (L;P)def() UP � UQ � Q � P .

char(k) = ` 6= p

(M ;Q) � (L;P)def() UP � UQ � Q � P .

char(k) = ` 6= p

(M ;Q) � (L;P)def() UP � UQ � Q � P .

char(k) = ` 6= p

(M ;Q) � (L;P)def() UP � UQ � Q � P .

char(k) = ` 6= p

(M ;Q) � (L;P)def() UP � UQ � Q � P .

G has a BN-pair

family of parabolic subgroups P � G :

char(k) = ` 6= p

(M ;Q) � (L;P)def() UP � UQ � Q � P .

char(k) = ` 6= p

(M ;Q) � (L;P)def() UP � UQ � Q � P .

char(k) = ` 6= p

(M ;Q) � (L;P)def() UP � UQ � Q � P .

LG = f(L;P) j P parabolic, L = P=UPg; char(k) = ` 6= p

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]

Y 2 Irrk(G ) is called LG -cuspidal if �RGL;P(Y ) = f0g when P 6= G .

L;P(X )o

Irrk(G ) =a

(L;X )=�

Irrk(G j (L;X )):

Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).

(L;P) 2 LG

and LL-cuspidal X 2 Irrk(L) Harish-Chandra series

L;P(X )o

Irrk(G ) =a

(L;X )=�

Irrk(G j (L;X )):

Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).

(L;P) 2 LG and LL-cuspidal X 2 Irrk(L)

Harish-Chandra series

L;P(X )o

Irrk(G ) =a

(L;X )=�

Irrk(G j (L;X )):

Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).

Irrk(G j (L;X )) :=nY 2 Irrk(G ) j

Y � RGL;P(X )

Irrk(G ) =a

(L;X )=�

Irrk(G j (L;X )):

Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).

L;P(X )o

Irrk(G ) =a

(L;X )=�

Irrk(G j (L;X )):

Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).

L;P(X )o

and corresponding Hecke algebra H(L;X )

:= EndkG (RGL;P(X )).

Irrk(G ) =a

(L;X )=�

Irrk(G j (L;X )):

Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).

L;P(X )o

Irrk(G ) =a

(L;X )=�

Irrk(G j (L;X )):

Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).

L;P(X )o

Irrk(G ) =a

(L;X )=�

Irrk(G j (L;X )):

Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).

L;P(X )o

If (L;P); (L;P 0) 2 LG ,

then RGL;P(X ) �= RG

Irrk(G ) =a

(L;X )=�

Irrk(G j (L;X )):

Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).

L;P(X )o

Irrk(G ) =a

(L;X )=�

Irrk(G j (L;X )):

Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).

L;P(X )o

Irrk(G ) =a

(L;X )=�

Irrk(G j (L;X )):

Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).

L;P(X )o

Irrk(G ) =a

(L;X )=�

Irrk(G j (L;X )):

Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).

L;P(X )o

Irrk(G ) =a

(L;X )=�

Irrk(G j (L;X )):

Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).

L;P(X )o

Irrk(G ) =a

(L;X )=�

Irrk(G j (L;X )):

Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).

L;P(X )o

Irrk(G ) =a

(L;X )=�

Irrk(G j (L;X )):

Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).

L;P(X )o

Irrk(G ) =a

(L;X )=�

Irrk(G j (L;X )):

Irrk(G j (L;X ))1�1 ! Irr(H(X ; L)).

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.

Complete results (classification of all LG -cuspidal simple modules and

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.

Irrk(G j (H ; kH))

= fV 2 Irrk(G ) j FixB(V ) 6= f0gg

k = IrrC(W ).)

H “cellular” in the sense of Graham–Lehrer,

cell modules

k = IrrC(W ).)

Irrk(G j (H ; kH))1�1 ! Irr(H)

has a natural parametrisation by

k = IrrC(W ).)

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

k = IrrC(W ).)

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].

Example. H has two 1-dimensional representations

ind : Ts 7! qcs

and " : Ts 7! �1 (s 2 S).

(If [G : B]1k 6= 0, then Λ�

StG = Steinberg module.

Simple , [G : B]1k 6= 0. [Steinberg 1957]

(If [G : B]1k 6= 0, then Λ�

Analogous result for classical groups G (q).

Description of Mullineux

(If [G : B]1k 6= 0, then Λ�

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

� � �

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

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

� � �

dV ;M =

multiplicity of M 2 Irrk(G ) in`-modular reduction of V 2 IrrK (G ).

�dV ;M

� � �

�dV ;M

� � �

General idea:

IrrK (G ) “known”, get hold of Irrk(G ) via D.

�dV ;M

� � �

General idea: IrrK (G ) “known”,

get hold of Irrk(G ) via D.

�dV ;M

� � �

�dV ;M

� � �

�dV ;M

� � �

�dV ;M

� � �

�dV ;M

� � �

�dV ;M

� � �

Each Di is square and det(Di ) = �1

�dV ;M

� � �

�dV ;M

� � �

�dV ;M

� � �

�dV ;M

� � �

�dV ;M

� � �

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.

D1 square matrix

) unipotent kG -modules in bijection with E1,

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?

D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.

Consider Lusztig’s a-function E1 ! N0, V 7! aV ,

and order E1

D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.

according to increasing a-invariant.

Then there is a unique bijection

D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.

Proved for GLn(q) [Dipper–James 1980/90s]

and GUn(q) [G. 1990].

D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.

Assume now G (q) = GLn(q) or GUn(q).

D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.

E1 = fV � j � ` ng,

ordered by reverse dominance order.

D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.

D1 = (d��)�;�`n

with d�� = 1 and d�� 6= 0 ) � E �.

D1 = (d��)�;�`n with d�� = 1

and d�� 6= 0 ) � E �.

D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.

Partition of fM� j � ` ng into HC-series?

When is M� cuspidal?

D1 = (d��)�;�`n with d�� = 1 and d�� 6= 0 ) � E �.

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.

i ni .

Gm�GL1(q2)(M�).

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 .

Gm�GL1(q2)(M�).

i ni .

Gm�GL1(q2)(M�).

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:

Gm�GL1(q2)(M�).

i ni .

Gm�GL1(q2)(M�).

i ni .

Gm�GL1(q2)(M�).

i ni .

Gm�GL1(q2)(M�).

i ni .

Gm�GL1(q2)(M�).

i ni .

Gm�GL1(q2)(M�).

i ni .

Gm�GL1(q2)(M�).

i ni .

Gm�GL1(q2)(M�).

i ni .

Gm�GL1(q2)(M�).

Then: Sources $ weakly cuspidal modules;

paths $ weak HC-series.

i ni .

Gm�GL1(q2)(M�).

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).

() ` not “linear”).

Then the HC-branching graph coincides with the

union of the crystal graphs of certain level 2 Fock spaces,

each of

M�;M� belong to the same weak HC-series of kGm-modules

shanghai, december 2015 - osaka universityariki/shanghai2015talkfiles/geck.pdf · this talk:...

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