Character sheaves and modular generalized Springer correspondence Part 2: The generalized Springer correspondence

Character sheaves and modular generalized Springer correspondence

Part 2: The generalized Springer correspondence

Anthony Henderson

(joint with Pramod Achar, Daniel Juteau, Simon Riche)

University of Sydney

January 2015

Simplifying the context

To get a ‘toy model’ of character sheaves onG:

1. Instead of G-equivariant perverse sheaves onG, consider G-equivariant perverse sheaves on the unipotent varietyU_G. This is simpler because there are only finitely many G-orbits, but still highly relevant e.g. for cuspidal character sheaves.

2. Assume p is large enough so that there is a G-equivariant isomorphismU_G → N^∼ _G where N_G is the nilpotent conein the Lie algebra g; then we can use Fourier transform on g.

3. The behaviour for large p is no different from considering G overC with the usual topology rather than ´etale topology.

This setting (first with simplification 1 only, later with 2 also) was studied by Lusztig in the case ofQ`-sheaves: one of his main results here was the ‘generalized Springer correspondence’.

Aim: to prove an analogue in themodular case wherechar(k) =`, as a first step towards understanding modular character sheaves.

The new set-up

New notation:

I G is a connected reductive algebraic group overC,

I g is its Lie algebra, on whichG has the adjoint action,

I N_G ={x ∈g|x nilpotent} is the nilpotent cone, on whichG has finitely many orbits,

I k is a sufficiently large field of characteristic`≥0,

I D_G(N_G,k) is the constructible equivariant derived category.

For anyA∈ D_G(N_G,k) andG-orbitO in N_G, the restrictions HⁱA|_O areG-equivariant local systems (i.e. G-equivariant sheaves of finite-dimensionalk-vector spaces) on O, so they correspond to finite-dimensional representations overk of the finite group

A_G(x) =G_x/G_x^◦, whereG_x is the stabilizer inG of x ∈ O.

LetN_G,k denote the set of pairs (O,E) whereO is a G-orbit in N_G andE is anirreducible G-equivariant local system onO.

Example (G =GL_n)

WhenG =GL_n,g=Mat_n andN_G ={x ∈Mat_n|xⁿ= 0}. By the Jordan form theorem, we have a bijection

G\N_G ←→ P_n={partitions λofn},

wherex∈ O_λ means that x has Jordan blocks of sizesλ1, λ2,· · ·. In this caseA_G(x) = 1 for all x, so N_G,k ←→ P_n for all fieldsk. Example (G of type G₂)

The five nilpotent orbits, in order of decreasing dimension, are:

G₂ (regular),G₂(a₁) (subregular), Af₁, A₁, 0.

These Bala–Carter labels record the type of the smallest Levi subalgebra meeting the orbit (whereAf₁ means the short-root A₁).

We haveAG(x) = 1 for all x exceptAG(x) =S3 for x∈G2(a1), so|N_G,k|= 7 usually,|N_G,k|= 6 ifchar(k)∈ {2,3}.

There is an anti-autoequivalenceD of D_G(N_G,k), Verdier duality.

We study the abelian subcategoryPerv_G(N_G,k) of G-equivariant perversek-sheaves on N_G, whereA∈ D_G(N_G,k) isperverseif

HⁱA|_O=Hⁱ(DA)|_O = 0 wheneveri >−dimO.

The simple objects inPerv_G(N_G,k) are in bijection withN_G,k: IC(O,E) = ‘intermediate extension’ of E[dimO] to O,

extended by zero to the whole of N_G.

Example (G =GL2, cf. Juteau–Mautner–Williamson)

The two orbits areO_(1,1)={0}and O₍₂₎=N_G \ {0}. We have IC(O_(1,1),k) =k₀ (skyscraper sheaf),

IC(O₍₂₎,k) =k_NG[2]if `6= 2.

The`= 2 case is different, because then H¹(O₍₂₎,k)6= 0.

Cuspidal pairs and induction series

LetP be a parabolic subgroup ofG andLa Levi factor of P. We have a geometric parabolic induction functor

I^G_L⊂P =Ind^G_P ◦Res^L_P :D_L(N_L,k)→ D_G(N_G,k), defined in the same way as for character sheaves:

Res^L_P :D_L(N_L,k)−→ D^∼ _P(N_L,k) ^(·)

∗

−→ D_P(N_P,k),

Ind^G_P :D_P(N_P,k)−→ D^∼ _G(G×_PN_P,k)−→ D^(·)! _G(N_G,k).

Lemma (Lusztig when` = 0, [AHR] when ` >0)

I^G_L⊂P commutes with D and maps Perv_L(N_L,k) to Perv_G(N_G,k).

It has left adjointR^G_L⊂P =Ind^L_P◦Res^G_P and right adjointR^G_L⊂P−

where P⁻ denotes the opposite parabolic with the same Levi L.

We say that a pair (O,E)∈N_G,k, or the correspondingIC(O,E), iscuspidalif the following equivalent conditions hold:

1. R^G_L⊂P(IC(O,E)) = 0 for allL⊂P (G;

2. IC(O,E) is not a quotient of I^G_L⊂P(A) for any L⊂P (G and any A∈Perv_L(N_L,k);

3. IC(O,E) is not a subobject of I^G_L⊂P(A) for any L⊂P (G and any A∈Perv_L(N_L,k).

Remark

When`= 0, the Decomposition Theorem of [BBD] implies that if A∈Perv<sub>L</sub>(N<sub>L</sub>,k) is simple, thenI<sup>G</sup><sub>L⊂P</sub>(A) is semisimple, so one can replace ‘quotient’/‘subobject’ with ‘summand’. Semisimplicity can fail if` >0, and cuspidalscan occur as constituents ofI^G_L⊂P(A).

This is analogous to modular representations ofG(Fq) when`6=p.

Lemma (Lusztig – same proof works for` >0)

If(O,E) is cuspidal,O is distinguished, i.e. meets no proper Levi.

LetM_G,k be the set ofcuspidal data (L,O_L,E_L) whereLis a Levi subgroup ofG (take only one representative of each G-conjugacy class, allowingL=G) and (O_L,E_L) is a cuspidal pair for L.

Proposition (Lusztig when`= 0, [AHJR] when ` >0) For any(L,O_L,E_L)∈M_G,k,I^G_L⊂P(IC(O_L,E_L))is independent of the parabolic P, and its head and socle are isomorphic.

Remark

The analogue for modular representations is by Geck–Hiss–Malle.

Theinduction seriesassociated to (L,O_L,E_L)∈M_G,k is the set of simple quotients (equivalently, subobjects) ofI^G_L⊂P(IC(O_L,E_L)).

Lemma (Lusztig – same proof works for` >0)

Any simple objectIC(O,E) in Perv_G(N_G,k)belongs to the induction series associated to some(L,O_L,E_L)∈M_G,k as above.

(IfIC(O,E) is cuspidal, then(L,O_L,E_L) = (G,O,E).)

The(modular) generalized Springer correspondenceis:

Theorem (Lusztig when` = 0, [AHJR] when ` >0)

1. Induction series associated to different cuspidal data are disjoint: in other words, a given IC(O,E) belongs to the induction series associated to a unique (L,O_L,E_L)∈M_G,k. 2. The induction series associated to (L,O_L,E_L)is canonically in

bijection with the set of irreducible k-reps of N_G(L)/L.

3. Hence we have a bijection

N_G,k ←→ G

(L,O_L,E_L)∈M_G,k

Irr(N_G(L)/L,k).

The proof will be discussed in the next lecture.

Remark

The analogue of 1 holds for modular representations ofG(Fq) also;

for the analogue of 2 one needs aq-deformed group algebra.

Background: the Springer correspondence

I In the mid-1970s, Springer gave a geometric construction of the irreducible Q`-reps of the Weyl groupW =NG(T)/T.

I As reformulated by Lusztig and Borho–Macpherson, this comes from an action ofW on the semisimple perverse sheaf

Spr=I^G_T⊂B(Q`0) =µ!Q`[dimN_G]∈PervG(N_G,Q`), whereµ:G×_B N_B → N_G is theSpringer resolutionof N_G. The Springer correspondenceis the resulting bijection

{simple summands ofSpr} ←→Irr(W,Q`) IC(O,E)7→Hom_Perv

G(N_G,Q`)(Spr,IC(O,E)).

I Lusztig then found that this was the (L,O_L,E_L) = (T,0,Q`) case of the generalized Springer correspondence, thus

accounting for the IC(O,E)’s that are not summands of Spr.

I Juteau (2007) showed that the Springer correspondence holds with k instead ofQ` and ‘quotients’ instead of ‘summands’.

Example (G =GL_n, W =S_n)

I When`= 0,|Irr(S_n,k)|=|P_n|, so everyIC(O_λ,k) is a summand of Spr, i.e. the Springer correspondence for GL_n is already ‘generalized’. In particular, GLn does not have a cuspidal pair unless n= 1.

I When` >0, James constructed the irreps D<sup>λ</sup> ofSn overk, labelled by λthat are `-regular (no part occurs≥`times).

Under Juteau’s correspondence,D^λ maps toIC(O_λt,k) where λ^t is the transpose partition; so these are the simple quotients of Spr. (All simples occur as constituents of Spr.)

The only distinguished orbit inN_G isO_(n); we will see that (O_(n),k) is cuspidal ⇐⇒ n is a power of`.

So M_G,k is essentially the set of Levis of the formQ

i≥0GL^m_`iⁱ, wherem_i are nonnegative integers such thatP

i≥0m_i`ⁱ =n.

Example (G =GL_n, ` >0 continued) ForL=Q

i≥0GL^m_`iⁱ such a Levi subgroup of GLn, we have N_G(L)/L∼=Y

i≥0

Smi, Irr(NG(L)/L,k)↔Y

i≥0

{`-regularλ<sup>(i)</sup>`mi}.

Under our correspondence, the collection (λ⁽ⁱ⁾) maps to IC(O_λ,k) whereλ=P

i≥0`ⁱ(λ⁽ⁱ⁾)^t. Note that IC(O_(n),k) occurs in the series ofL=Q

i≥0GL^b_`iⁱ whereP

i≥0bi`<sup>i</sup> =n and all bi < `.

Remark

The above combinatorial correspondence is a simplified version of what appears in the analogous theory of induction series for modular representations ofGLn(Fq) (Dipper–Du).

Example (G =G₂, W dihedral of order 12)

I When`= 0,|Irr(W,k)|= 6<7 =|N_G,k|. The non-Springer pair is (G2(a1),E_sign), which must be cuspidal because the other proper Levi subgroups are both isomorphic toGL₂.

I When`= 2,|Irr(W,k)|= 2, and onlyIC(0,k) andIC(Af1,k) belong to Juteau’s correspondence. The other series are:

(L of typeA1,O₍₂₎,k), |N_G(L)/L|= 2 : IC(A1,k),

(L of typeAf₁,O₍₂₎,k), |N_G(L)/L|= 2 : IC(G₂(a₁),E_refln), leaving 2 cuspidal pairs, (G₂(a₁),k) and (G₂,k).

The aboveGL_n andG₂ examples illustrate:

Theorem ([AHJR])

When` >0,IC(O<sub>reg</sub>,k) belongs to the induction series associated to(L,O<sub>L,reg</sub>,k)where L is minimal such that `-|W/W_L|.