Local group theory : from Frobenius to Derived Categories

Local group theory : from Frobenius to Derived Categories

Michel Brou´e

Universit´e Paris–Diderot Paris 7

September 2012

LOCAL GROUP THEORY

Feit–Thompson, 1963

If G is a non abelian simple finite group, then 2 | | G | .

Cauchy (1789–1857)

If p | | G | , there are non trivial p –subgroups in G . Sylow, 1872

The maximal p–subgroups of G are all conjugate under G .

Brauer–Fowler, 1956

There are only a finite number of isomorphism types of finite simple

groups with a prescribed type of centralizer of an involution.

Assume P ⊂ S and P ⊂ S

. There is g ∈ G such that S

= S

(= g

Sg ), hence

P ⊂ S and

P (= gPg

) ⊂ S . This is a fusion.

The Frobenius Category Frob

(G ) :

Objects : the p–subgroups of G,

Hom(P, Q) := { g ∈ G | (

P ⊂ Q) } /C

(P) .

Note that Aut(P ) = N

(P )/C

(P ) .

Alperin’s fusion theorem (1967) says essentially that Frob

(G ) is

known as soon as the automorphisms of some of its objects are

known.

Main question of local group theory

How much is known about G from the knowledge (up to equivalence of categories) of Frob

(G ) ?

Well, certainly not more than G /O

(G ) !

(where O

(G ) denotes the largest normal subgroup of G of order not divisible by p)

Indeed, O

(G ) disappears in the Frobenius category, since, for P a p–subgroup,

O

(G ) ∩ N

(P ) ⊆ C

(P ) .

But perhaps all of G /O

(G ) ?

Control subgroup

Let H be a subgroup of G . The following conditions are equivalent : (i) The inclusion H ֒ → G induces an equivalence of categories

Frob

(H) −→

Frob

(G ) ,

(ii) H contains a Sylow p –subgroup of G , and if P is a p–subgroup of H and g is an element of G such that

P ⊆ H, then there is h ∈ H and z ∈ C

(P ) such that g = hz.

If the preceding conditions are satisfied, we say that H controls

p–fusion in G , or that H is a control subgroup in G .

The first question may now be reformulated as follows :

If H controls p–fusion in G , does the inclusion H ֒ → G induce an isomorphism

H/O

(H) −→

G /O

(G ) ? In other words, do we have

G = HO

(G ) ?

Frobenius theorem, 1905

If a Sylow p–subgroup S of G controls p–fusion in G , then the inclusion induces an isomorphism S ≃ G/O

(G ).

p–solvable groups, ?

Assume that G is p–solvable. If H controls p-fusion in G , then the inclusion induces an isomorphism H/O

(H) ≃ G /O

(G ).

Z

–theorem (Glauberman, 1966 for p = 2, Classification for other primes)

Assume that H = C

(P ) where P is a p–subgroup of G . If H controls p-fusion in G , then the inclusion induces an

isomorphism H/O

(H) ≃ G /O

(G ).

But

Burnside (1852–1927)

Assume that a Sylow p –subgroup S of G is abelian. Set

H := N

(S ). Then H controls p-fusion in G .

Consider the Monster , a finite simple group of order

2

· 3

· 5

· 7

· 11

· 13

· 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 ≃ 8.10

. (predicted in 1973 by Fischer and Griess, constructed in 1980 by Griess, proved to be unique by Thompson)

and the normalizer H of one of its Sylow 11–subgroups, a group of order 72600, isomorphic to (C

× C

) ⋊ (C

× SL

(5)) (here we denote by C

the cyclic group of order m).

Here we have G 6 = HO

(G ) since G is simple.

Remark : one may think of more elementary examples...

LOCAL REPRESENTATION THEORY

Let K be a finite extension of the field of p –adic numbers Q

which contains the | G | -th roots of unity. Let O be the ring of integers of K over Z

, with maximal ideal p and residue field k := O /p.

K 2 R O

dd ■■

■■ // // k = O /p Q

?

OO

Z

1 Q

cc ●● ● ?

OO // // F

= Z

/pZ

^?

OO

Block decomposition

O G = L

B (indecomposable algebra)

↓ ↓

kG = L

kB (indecomposable algebra)

The augmentation map O G → O factorizes through a unique block of O G called the principal block and denoted by B

(G ).

O G ● ● ● ● ● ● ● ● // ● B ● ##

(G )

O

Set e

(G ) :=

p′(G)|

P

s∈Op′(G)

s. Then e

(G ) is a central idempotent of

O G and O Ge

(G ) is a product of blocks containing the principal block

B

(G ).

Factorisation and principal block

If H is a subgroup of G , the following assertions are equivalent (i) G = HO

(G) .

(ii) The map Res

induces an isomorphism from O Ge

(G ) onto O He

(H).

In particular, in that case, the map Res

induces an isomorphism from B

(G ) onto B

(H).

== Let us re-examine the counterexamples to factorization coming from Burnside’s theorem.

Assume that a Sylow p–subgroup S of G is abelian, let H := N

(S) be its normalizer.

Even if G 6 = H O

(G), it appears that there is some connection

between the (representation theory of) B

(G ) and B

(H).

First of all, there are many examples where there is no factorization, but where the algebras are Morita equivalent — but then not through the Res

functor.

A kind of generic example :

G = GL

(q )

B H

❙❙❙ ❙❙❙

❙❙❙ ❙❙❙

❙❙❙ ❙❙❙ ❙❙

U

q q q q q q

q q T = S × T

❙❙❙ ❙❙❙ ❙❙❙ ❙

q q q q q q q

1

▼▼ ▼▼ ▼▼

▼▼ ▼

❦ ❦

❦ ❦

❦ ❦

❦ ❦

❦ ❦

p ∤ q , p > n, S p-Sylow

H := N

(T ) = N

(S)

←− H/T = S

We certainly have

G 6 = HO

(G ) .

But the principal block algebras of G and H respectively are Morita equivalent.

There exist M and N, respectively a O G –module– O H and a O H–module– O G with the following properties :

(With appropriate cutting by the principal block idempotents) M ⊗

N ≃ B

(G ) as O G –modules– O G N ⊗

M ≃ B

(H) as O H–modules– O H Viewed as an O G –module– O T , we have

M ≃ O (G /U ) ,

i.e., the functor M ⊗

? is the Harish–Chandra induction.

SOME NUMERICAL MIRACLES

Let us consider the case G = A

and p = 2. Then we have H ≃ A

. Remark : on a larger screen, we might as well consider the above case of the Monster and of the prime p = 11.

Table : Character table of A

(1) (2) (3) (5) (5’)

1 1 1 1 1 1

χ

4 0 1 -1 -1

χ

5 1 -1 0 0

χ

3 -1 0 (1 + √

5)/2 (1 − √ 5)/2

χ

3 -1 0 (1 − √

5)/2 (1 + √

5)/2

Table : Character table of B

(A

)

(1) (2) (5) (5’) (3)

1 1 1 1 1 1

χ

5 1 0 0 -1

χ

3 -1 (1 + √

5)/2 (1 − √

5)/2 0

χ

3 -1 (1 − √

5)/2 (1 + √

5)/2 0

Table : Character table of A

(1) (2) (3) (3’)

1 1 1 1 1

− α

-3 1 0 0

− α

-1 -1 (1 + √

− 3)/2 (1 − √

− 3)/2

− α

-1 -1 (1 − √

− 3)/2 (1 + √

− 3)/2

ABELIAN SYLOW CONJECTURE

Assume that a Sylow p–subgroup S of G is abelian, let H := N

(S) be its normalizer.

(ASC) :

The algebras B

(G ) and B

(H) are derived equivalent.

Which means :

There exist M and N, respectively a bounded complex of B

(G )–modules–B

(H) and a bounded complex of B

(H)–modules–B

(G ) with the following properties :

M ⊗

N ≃ B

(G ) as complexes of B

(G )–modules–B

(G )

N ⊗

M ≃ B

(H) as complexes of B

(H)–module–B

(H)

(Strong ASC) :

They are Rickard equivalent, that is, derived equivalent in a way which is compatible with the equivalence of Frobenius

categories

Which means : There is a G –equivariant collection of derived equivalences

{ E (P) : D

(B

(C

(P)) −→ D

(B

(C

(P )) }

compatible with Brauer morphisms.

RICKARD’S EXPLANATION FOR A

G := A

H := N

(S

), (S

a Sylow 2–subgroup of G ) View B

(G ) as acted on as follows

B2(G)

B

(G )

2(H)

B2(G)

IB

(G )

2(H)

:= kernel of augmentation map B

(G ) ։ O .

C := a projective cover of the bimodule IB

(G ).

Thus we have

C

$$ ■ ■ ■ ■ ■

{0}

//

//

B₂(G)

//

{0}

set

Γ

:= { 0 } → C → B

(G ) → { 0 }

where B

(G ) is in degree 0 (and C in degree − 1).

We have homotopy equivalences : Γ

⊗

OH

Γ

≃ B

(G ) as complexes of (B

(G ), B

(G ))–bimodules, Γ

⊗

OG

Γ

≃ B

(H) as complexes of (B

(H), B

(H))–bimodules.

Case of finite reductive groups

Usual notation

G is a connected reductive algebraic group over F

, with Weyl group W , endowed with a Frobenius endomorphism F defining a rational structure over F

.

Here we assume that ( G , F ) is split.

G := G

is the corresponding finite reductive group, with order

| G | = q

Y

d>0

Φ

(q)

a polynomial which depends only on the reflection representation of W on Q ⊗ Y ( T ).

Indeed, that polynomial is

q

Y

i

(q

− 1) .

Sylow Φ

–subgroups, d –cyclotomic Weyl group

There exists a rational torus S

of G , unique up to G –conjugation, such that

| S

| = | S

d

| = Φ

(q)

. Set L

:= C

( S

) and N

:= N

( S

) = N

( L

)

W

:= N

/L

is a true finite group, a complex reflection group in its action on C ⊗ Y ( S

).

= This is the d –cyclotomic Weyl group of the finite reductive group G .

Example : For G = GL

(q ) and n = dm + r (r < d ), then

L

= GL

(q

)

× GL

(q) , W

= µ

≀ S

The Sylow ℓ–subgroups and their normalizers ℓ a prime number, prime to q, ℓ | | G | , ℓ ∤ | W |

= ⇒ There exists one d (a(d ) > 0) such that ℓ | Φ

(q), and the Sylow ℓ–subgroup S

of S

is a Sylow of G .

L

= C

(S

) and N

= N

= N

(S

) : N

L

1 Since the “local” block is

B

(Z

N

) ≃ Z

[S

⋊ W

] our conjecture reduces to

Conjecture

D

(B

(Z

G )) ≃ D

(Z

[S

⋊ W

])

Role of Deligne–Lusztig varieties

Let P be a parabolic subgroup with Levi subgroup L

, and with unipotent radical U .

Note that P is never rational if d 6 = 1.

The Deligne–Lusztig variety is V

:=

G

{ g U ∈ G / U | g U ∩ F (g U ) 6 = ∅}

L

hence defines an object

RΓ

( V

, Z

) ∈ D

(

mod

ℓLd

) Conjecture

There is a choice of U such that

1

RΓ

( V

, Z

)

is a Rickard complex between B

(Z

G ) and its

commuting algebra C ( U ).

The case where d = 1

If d = 1,

S

= T = L

and W

= W

V

= G/U and RΓ

( V

, Z

) = Z

(G /U )

Z

G

Z

(G /U )

C (U) where

1

C (U ) ≃ Z

T .Z

H (W , q)

2

Q

H (W , q) ≃ Q

W

The unipotent part

Extend the scalar to Q

=: K ⇒ Get into a semisimple situation RΓ

( V ( U ), Z

) becomes

H

( V ( U ), K ) := M

i

H

( V ( U ), K )

Replace V ( U ) by V ( U )

:= V ( U )/L

⇒

Only unipotent characters of G are involved Semisimplified unipotent conjecture

1

The different H

( V ( U )

, K ) are disjoint as KG–modules,

2

H ( U ) := End

H

( V ( U )

, K ) ≃ KW

Case where d is regular

L

=: T

is a torus ⇐⇒ d is a regular number for W The set of tori L

is a single orbit of rational maximal tori under G , hence corresponds to a conjugacy class of W .

For w in that class, we have W

≃ C

(w ).

The choice of U corresponds to the choice of an element w in that class.

We then have

V ( U

)

= X

:= { B ∈ B | B →

F ( B ) } B is the variety of all Borel subgroups of G

The orbits of G on B × B are canonically in bijection with W

and we write B →

B

if the orbit of ( B , B

) corresponds to w .

Relevance of the braid groups

Notation

V := C ⊗ Y ( T ) acted on by W , A := set of reflecting hyperplanes of W V

:= V − S

H∈A

H B

:= Π

(V

/W , x

)

“Section” W → B

, w 7→ w , since If W =< S | ststs . . .

| {z }

ms,t factors

= tstst . . .

| {z }

ms,t factors

, s

= t

= 1 >

then B

=< S | ststs . . .

| {z }

ms,t factors

= tstst . . .

| {z }

ms,t factors

>

π := t 7→ e

x

= ⇒ π ∈ ZB

2 h

A theorem of Deligne

Theorem (Deligne)

Whenever b ∈ B

there is a well defined variety X

b

) such that X

= X

.

For b = w

w

· · · w

we have X

b

= { ( B

, B

, . . . , B

) | B

→

B

→ · · ·

→

B

and B

= F ( B

) }

The variety X

X

= { ( B

, B

, B

) | B

→

B

→

B

and B

= F ( B

}

= { ( B

, B

, . . . , B

) | B

→

B

→ · · ·

→

B

and B

= F ( B

) } The (opposite) monoid B

acts on X

: For w ∈ B

, and

π = w b = b w ,

if B →

B

→

F (B )

D

: ( B , B

, B

= F ( B )) 7→ ( B

, F ( B ), F ( B

)) Hence B

acts on H

( X

)

Proposition : The action of B

on H

( X

) factorizes through the (ordinary) Hecke algebra H (W ).

Conjecture :

End

H

( X

) = H (W )

Relevance of roots of π

Proposition

d regular for W ⇐⇒ there exists w ∈ B

such that w

= π.

Application

1

X

embeds into X

:

X

֒ → X

B 7→ ( B , F ( B ), . . . , F

( B ))

2

Its image is

{ x ∈ X

| D

( x ) = F ( x ) }

3

C

W

( w ) acts on X

.

Belief

A good choice for U

is : w a d –th root of π.

Theorem (David Bessis) There is a natural isomorphism

B

−→

C

( w ) From which follows :

Theorem

The braid group B

of the complex reflections group C

(w ) acts on H

( X

).

Conjecture

The braid group B

acts on H

( X

) through a d –cyclotomic Hecke

algebra H

(w ).

d –cyclotomic Hecke algebras

A d –cyclotomic Hecke algebra for C

(w ) is in particular an image of the group algebra of the braid group B

,

a deformation in one parameter q of the group algebra of C

(w ), which specializes to that group algebra when q becomes e

Examples :

The ordinary Hecke algebra H (W ) is 1–cyclotomic, Case where W = S

, d = 3 :

C

(w ) = B

(3) = µ

≀ S

←→

s

3

t 2

H

(w ) =

S , T ;



 

 

STST = TSTS

(S − 1)(S − q)(S − q

) = 0 (T − q

)(T + 1) = 0



 

 

W = D

, d = 4, C

(w ) = G (4, 2, 2) ←→

♥

H

(w ) =

S, T , U ;

( STU = TUS = UST (S − q

)(S − 1) = 0

)

Let us summarize

1

ℓ d , d regular, i.e., L

= T

, w

= π, V ( U

)/L

= X

2

End

H

( X

) ≃ H

(w )

3

Z

H

(w ) −→

Z

C

(w )

4

End

RΓ

( V ( U

), Z

) ≃ Z

(T

)

· End

RΓ

( X

, Z

) ≃ B

(Z

N

)

What is really proven today

Everything

if d = 1 (Puig),

for G = GL

(q) (Rouquier), SL

(q) (cf. a book by Bonnaf´e) for G = GL

(q) and d = n (Bonnaf´e–Rouquier)

About : End

H

( X

) ≃ H

(w ) ? All H

(w ) are known, all cases (Malle) Assertion End

H

( X

) ≃ H

(w ) known for

d =h(Lusztig),

d = 2 (Lusztig, Digne–Michel), small rank GL,

d = 4 forD4(q) (Digne–Michel).