SLIDE 1 Functions
◮ Our style this and next week is going to change: the goal is to talk about connection between representation theory of finite reductive groups and perverse sheaves. ◮ From now on let us work with Qℓ-sheaves on the (small) ´ etale site. ◮ Let us begin with function-sheaf dictionary: Given X0 a variety over Fq and F ∈ D(X0), one consider for any x ∈ X0(Fq): f F(x) :=
(−1)k Tr(Fr∗; Hk(F)¯
x) ∈ Qℓ
where ¯ x is the Fq-point above x. ◮ We have
- 1. f F = f F′ + f F′′ for F ′ → F → F ′′
+1
− →.
- 2. f F⊗G = f F · f G.
- 3. f F[d] = (−1)df F.
- 4. For a morphism g : Y0 → X0, we have f g∗F = g ∗f F.
- 5. For a morphism h : X0 → Y0, we have f h!F = h!f F, where
g! : C(X0(Fq)) → C(Y0(Fq)) is defined by h!f (y) =
x∈(h−1(y))(Fq) f (x).
SLIDE 2 Function-sheaf dictionary
◮ One can likewise define f F
m : X0(Fqm) → Qℓ as f Fm where Fm is the
pullback of F to (the ´ etale site of) X0 ×Spec Fq Spec Fqm. One has ◮ Proposition. For F1, F2 ∈ Perv(X0) semisimple we have F1 ∼ = F2 iff f F1
m
= f F2
m
for any m ∈ Z≥1. ◮ Since perverse sheaves are determined by a local system on a locally closed subset U, this reduces to local system on U which is then given by a Chebotarev density theorem for representations of π´
et 1 (U).
◮ Nevertheless, we have that F ∈ Perv(X0) gives rise to ˜ F ∈ Perv(X) with Fr∗ ˜ F ∼ = ˜
F is known to be simple, then this isomorphism is unique up to Q
× ℓ . So one may work with ˜
F ∈ Perv(X) and knowing that associated functions f F
m are unique up to some
constant C m.
SLIDE 3 Induction
◮ Suppose one wants to study the representation theory of G(Fq) for some split reductive group G where you might take the case G = GLn for some fixed n. A standard consideration is to begin with T ⊂ B ⊂ G where T is the (algebraic) diagonal torus and B is the closed subgroup of upper triangular elements. ◮ There is an algebraic surjection B ։ T which induces B(Fq) ։ T(Fq). ◮ One consider θ : T(Fq) → Qℓ some character, pull it back to B(Fq), then induce it to ˜ theta = θ ⊗Qℓ[B(Fq)] Qℓ[G(Fq)]. ◮ As a finite-dimensional representation of G(Fq), its character is given by ˜ θ(g) =
θ(h−1gh), ∀g ∈ G(Fq) where θ is considered to be zero on G(Fq) − B(Fq).
SLIDE 4 Sheaf-theoretic induction
˜ θ(g) =
θ(h−1gh), ∀g ∈ G(Fq).
◮ Note that NG(B) = B, and also G(Fq)/B(Fq) = (G/B)(Fq) since by Lang’s theorem H1(Fq; B) is trivial. Hence G/B (both rationally and algebraically) may and shall be identified with the flag variety: the variety of conjugates of B in G. When G = GLn, this is also the variety of full flags in our n-dimensional standard representation. ◮ Consider ˜ G := {(g, B) ∈ G × G/B | g ∈ B}. = {(g, hB) ∈ G × G/B | h−1gh ∈ B} ◮ We have two natural maps π : ˜ G ։ G by sending (g, hB) to g and ρ : ˜ G ։ T by sending h−1gh ∈ B to its image in T. ◮ There is a general way to associate characters on T(Fq) to sheaves
- n T. For the moment, let us look at the baby case θ = 1 and we
take the constant sheaf QℓT on T. Consider K := π!ρ∗QℓT. ◮ Claim. We have f K = ˜ θ as functions on G(Fq).
SLIDE 5 Springer theory
˜ G := {(g, hB) ∈ G × G/B | h−1gh ∈ B}.
◮ Now the highlight is: ◮ Proposition. We have K[dim G] := π!ρ∗QℓT[dim G] ∈ Perv(G). It is semisimple, with components in bijection with components of the regular representation of W . Here W = Sn when G = GLn. ◮ Proof. The only highlight is that ˜ G → G is small. Once we have this, π!ρ∗QℓT is determined by π on a sufficiently small Zariski open
- f G, which one might take to be G rs the open subvariety of regular
semisimple elements; for G = GLn this is those with distinct
- eigenvalues. Then π−1G rs → G rs is finite ´
etale with Galois group W (= Sn if G = GLn). Hence we have π´
et 1 (G rs) → W and π!Qℓπ−1G rs
factors through the regular representation of W . Since K (up to shift) is the middle extension of π!Qℓπ−1G rs, the result follows. ◮ In fact, for each ρ ∈ Irr(W ) (irreducible Qℓ-representations of W ) let Fρ ∈ Perv(G) be the corresponding simple object, then we have a Qℓ[W ]-equivariant isomorphism: K[dim G] ∼ =
Fρ ⊗ ρ where the latter ρ is treated as a Qℓ[W ]-module.
SLIDE 6
Smallness
◮ Proposition. π : ˜ G → G is small. ◮ Proof. To say π is semi-small (given the evident properness) is equivalent to saying that dim ˜ G ×G ˜ G = dim G. ◮ Recall ˜ G = {(g, B′) ∈ G × G/B | g ∈ B′} where we view G/B as the variety of conjugates of B. Hence ˜ G ×G ˜ G = {(g, B′, B′′) ∈ G × G/B × G/B | g ∈ B′ ∩ B′′}. ◮ The G-orbit of (B′, B′′) ∈ G/B × G/B is the so-called relative position of B′ and B′′; we have G\(G/B × G/B) ∼ = B\G/B ∼ = W . So there is a stratification on ˜ G ×G ˜ G indexed by W where each stratum maps to the corresponding G-orbit in G/B × G/B. ◮ The orbit in G/B × G/B indexed by w has dimension = dim G/B + l(w) and the fiber above the orbit has dimension dim(B ∩ wB) = dim B − l(w). Hence the strata always have dimension = dim G. ◮ All such stratum has dense image in G as B ∩ wB ⊃ T and conjugates of T are dense in G. Hence each stratum of dim ˜ G ×G ˜ G has dense image in G, and π is not only semi-small but also small.
SLIDE 7 Decomposition of K
K[dim G] := π!ρ∗QℓT [dim G] ∈ Perv(G) is semisimple, with components in bijection with components of the regular representation of W .
◮ Recall that f K is the character of ˜ 1 := 1 ⊗Qℓ[B(Fq)] Qℓ[G(Fq)]. ◮ We also have EndQℓ[G(Fq)](˜ θ) ∼ =
- w∈B(Fq)\G(Fq)/B(Fq) HomB(Fq)∩wB(Fq)(θ, wθ) as
vector spaces. In particular EndQℓ[G(Fq)](˜ 1) ∼ = Qℓ[W ] as vector spaces. ◮ The last isomorphism can be proved to be an algebra isomorphism. ◮ It is thus natural to expect irreducible components of EndQℓ[G(Fq)](˜ 1) are in bijections with components of K (or K[dim G] ∈ Perv(G)) under function-sheaf dictionary. ◮ Proposition. This is not true in general, but true for G = GLn. ◮ Sketch. When G = GLn, W = Sn. Class functions of Sn can be generated by inductions from trivial representation of Sm1 × ... × Smk ⊂ Sn with m1 + ... + mk = n. Such a subgroup corresponds to P ⊂ GLn the group of blockwise upper triangular matrices with blocks of size mi. And the induction corresponds to a KP given by replacing B by P in the construction of K.
SLIDE 8 Another point of view
◮ Let g = Lie G, and ˜ g = {(X, hB) ∈ g × G/B | h−1Xh ∈ Lie B}. Denote by πg := ˜ g → g the natural map. ◮ Consider N ⊂ g to be the closed subvariety of nilpotent elements, and ˜ N := π−1
g (N). Write πN : ˜
N → N. We also have ◮ Proposition πg is small and πN is semi-small. ◮ The proof is the same; there are again strata indexed by W of the same dimension in ˜ g ×g ˜ g and ˜ N ×N ˜
- N. The only difference is that
stratum in ˜ N ×N ˜ N no longer has dense image to N. ◮ Write U to be the unipotent radical of B; strictly upper triangular matrices inside upper triangular matrices if G = GLn, and u = Lie U. We have ˜ N = {(X, hB) | h−1Xh ∈ u}, and each stratum of ˜ N ×N ˜ N has image being conjugates of u ∩ wu. ◮ Example G = GL3. Then the stratum indexed by id ∈ S3 has image being conjugates of u, namely the whole N. The stratum indexed by (321) ∈ S3 has image in u ∩ wu = 0. All the rest has images in conjugates of 0 u ∩ wu u, which is Nsub ⊂ N consisting of nilpotent elements whose Jordan blocks have size ≤ 2.
SLIDE 9 Springer theory II
◮ What we saw is that in the case G = GL3, the map πN : ˜ N → N has πN Qℓ ˜
N [dim N] ∈ Perv(N) and (thanks to decomposition
theorem) its image has some components supported on 0 and on Nsub. ◮ And it very much looks like, though unclear how, that these components have to do with W . ◮ Write i : N ֒ → g, Kg := πgQℓ˜
g[dim G] and
KN = (πN )∗Qℓ ˜
N [dim N]. We have i∗[dim N − dim G]Kg = KN by
proper base change. Recall that W acts on Kg and thus induces a W -action on KN . ◮ Theorem. (Borho-MacPherson) Let i : N ֒ → g. The functor i∗[dim N − dim G] takes simple sub-objects of KG to simple sub-objects of KN . ◮ In particular, each simple sub-object of KN ∈ Perv(N) corresponds to an irreducible representation of W . All such sub-object are necessarily G-conjugation invariant, and thus comes from a G-conjugation orbit on N and a local system on the orbit.
SLIDE 10 Springer theory III
Each simple sub-object of KN ∈ Perv(N) corresponds to an irreducible representation of W . All such sub-object are necessarily G-conjugation invariant, and thus comes from a G-conjugation
- rbit on N and a local system on the orbit.
◮ In fact, for ρ ∈ Irr(W ) let us denote by Fρ the corresponding simple sub-object of KN , then we have a Qℓ[W ]-equivariant isomorphism: KN ∼ =
Fρ ⊗ ρ where the latter ρ is again treated as a Qℓ[W ]-module. ◮ For example, when G = GL3, F1 is the shifted constant sheaf on N. Fsgn is the skyscraper sheaf supported at 0 ∈ N. The last 2-dimensional representation η ∈ Irr(S3) corresponds to Fη = ICNsub ∈ Perv(N). ◮ In fact, since (KN )0 = H∗(G/B)[6] = Qℓ[6] ⊕ Qℓ[4]⊗2 ⊕ Qℓ[2]⊗2 ⊕ Qℓ. We can even use this to compute (ICNsub)0 = (j!∗QℓNsub−{0}[4])0 = Qℓ[4] ⊕ Qℓ[2].
SLIDE 11
Equivariant perverse sheaves
◮ The map ρ → Fρ is called Springer correspondence, but we like it from from Irr(W ) to PervG(N). ◮ The notion of G-equivariant perverse sheaves goes through quite some complication. A G-equivariant sheaf (underived) is (F, φ) with F ∈ Sh(X) and φ : a∗F ∼ = p∗F satisfying natural conditions, where a : G × X → X is the action map and p2 : G × X → X is the projection to the 2nd factor. Denote by ShG(X) the category of G-equivariant sheaves on X. ◮ Example G a smooth connected group and X = pt. Then ShG(pt) = Sh(pt). ◮ This is not a good thing to be derived; the derived category DG(X) can’t/shouldn’t be constructed directly from ShG(X). ◮ Now assume that G is smooth and connected. It happens that after a bunch of complications, PervG(X) can be defined as the full subcategories of F ∈ Perv(X) with a∗F ∼ = p∗F (isomorphism in Perv(X)).
SLIDE 12 Cuspidality
◮ We want to view ρ → Fρ ∈ PervG(N) as “Fρ lives in the principal series.” ◮ For any parabolic subgroup iP : P ֒ → G, write πP : P ։ P/UP the Levi quotient. Under the Harish-Chandra philosophy, we say F ∈ PervG(N) is cuspidal if πP ∗ i!
PF = 0.
◮ Suppose L ֒ → P ֒ → G is a Levi and parabolic subgroup, we consider ˜ NP = {(X, hP) ∈ N × G/P | h−1Xh ∈ Lie P ∩ N} with natural map π : ˜ NP → ˜ N and the other map λ : ˜ N → [NL/L] by (X, hP) → h−1Xh. ◮ Lusztig proved that we have a partition PervG(N) =
perverse components of π!λ∗F. =
Irr(NG(L)/L). The principal series corresponds to the case L = T, F trivial and we get Irr(W ) parameterizing the components.
SLIDE 13 Character sheaves on a torus
◮ We wish to end our topic by briefly talking about character sheaves; sheaves that we use to understand all characters on G(Fq) after Lusztig. ◮ To describe the result we need the case of torus. Let T be a torus
- ver Fq. Then T(Fq) = T(Fq)F = (X∗(T) ⊗ F
× q )F where F is the
arithmetic Frobenius which also acts on X∗(T) = Hom(Gm, T). ◮ We may choose a F-equivariant embedding F
× q ֒
→ Q/Z, where F acts on Q and Z by multiplication by q. The cokernel is p-power torsion, which implies T(Fq) ∼ = (X∗(T) ⊗ (Q/Z))F. ◮ We have X∗(T) ⊗ Z X∗(T) ⊗ Q X∗(T) ⊗ (Q/Z) X∗(T) ⊗ Z X∗(T) ⊗ Q X∗(T) ⊗ (Q/Z)
F−1 F−1 F−1
◮ Snake lemma gives T(Fq) ∼ = (X∗(T) ⊗ (Q/Z))F ∼ = (X∗(T) ⊗ Z)F, and Hom(T(Fq), Q
× ℓ ) ∼
= (X ∗(T) ⊗ Q/Z)F.
SLIDE 14 From T to G
Hom(T(Fq), Q
× ℓ ) ∼
= (X ∗(T) ⊗ Q/Z)F .
◮ Hence a character θ ∈ Hom(T(Fq), Q
× ℓ ) gives χθ ∈ X ∗(T) ⊗ Q/Z
with prime-to-p order. Any such element gives an element in Hom(π´
et 1 (T × Spec Fq), Q × ℓ ).
◮ Lemma. Any character θ ∈ Hom(T(Fq), Q
× ℓ ) ∼
= (X ∗(T) ⊗ Q/Z)F is the function associated to a local system Lθ on (the ´ etale site of) T given by χθ. ◮ Now let us base change all to Fq. Recall we had ˜ G G T
π ρ
where ˜ G = {(g, hB) ∈ G × G/B | h−1gh ∈ B} and ρ(g, hB) = h−1gh. Examples of character sheaves are then perverse constituents of π!ρ∗Lθ.
SLIDE 15
Character sheaves
◮ More generally, for any w ∈ W , we may consider ˜ Gw = {(g, hB) ∈ G × G/B | h−1gh ∈ BwB} Now ˜ Gw does not have a map to T - because of the obstruction of the conjugation by w. It can be shown that if Lθ is invariant by w, then “pullback of Lθ to ˜ Gw” still make sense. ◮ Let’s not go into that full detail, but just call Fθ,w the resulting “pullback” local system on ˜ Gw. ◮ Definition-Theorem. (Lusztig) Character sheaves on defined to be perverse constituents of π!Fθ,w for all w ∈ W and w-invariant θ. All irreducible characters on G(Fq) can be expressed as a linear combination of functions associated to character sheaves, in an explicit manner that is independent of q.
