Functions Our style this and next week is going to change: the goal - PowerPoint PPT Presentation

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 Q ℓ -sheaves on the (small) ´ etale site. ◮ Let us begin with function-sheaf dictionary: Given X 0 a variety over F q and F ∈ D ( X 0 ), one consider for any x ∈ X 0 ( F q ): ( − 1) k Tr(Fr ∗ ; H k ( F ) ¯ f F ( x ) := � x ) ∈ Q ℓ k where ¯ x is the F q -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) d f F . 4. For a morphism g : Y 0 → X 0 , we have f g ∗ F = g ∗ f F . 5. For a morphism h : X 0 → Y 0 , we have f h ! F = h ! f F , where g ! : C ( X 0 ( F q )) → C ( Y 0 ( F q )) is defined by h ! f ( y ) = � x ∈ ( h − 1 ( y ))( F q ) f ( x ).

Function-sheaf dictionary m : X 0 ( F q m ) → Q ℓ as f F m where F m is the ◮ One can likewise define f F pullback of F to (the ´ etale site of) X 0 × Spec F q Spec F q m . One has ◮ Proposition. For F 1 , F 2 ∈ Perv( X 0 ) semisimple we have F 1 ∼ = F 2 iff f F 1 = f F 2 for any m ∈ Z ≥ 1 . m m ◮ 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( X 0 ) gives rise to ˜ F ∈ Perv( X ) with Fr ∗ ˜ F ∼ = ˜ F . If ˜ F is known to be simple, then this isomorphism × ℓ . So one may work with ˜ F ∈ Perv( X ) and is unique up to Q knowing that associated functions f F m are unique up to some constant C m .

Induction ◮ Suppose one wants to study the representation theory of G ( F q ) for some split reductive group G where you might take the case G = GL n 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 ( F q ) ։ T ( F q ). ◮ One consider θ : T ( F q ) → Q ℓ some character, pull it back to B ( F q ), ˜ theta = θ ⊗ Q ℓ [ B ( F q )] Q ℓ [ G ( F q )]. then induce it to ◮ As a finite-dimensional representation of G ( F q ), its character is given by ˜ � θ ( h − 1 gh ) , ∀ g ∈ G ( F q ) θ ( g ) = h ∈ G ( F q ) / B ( F q ) where θ is considered to be zero on G ( F q ) − B ( F q ).

Sheaf-theoretic induction ˜ θ ( h − 1 gh ) , ∀ g ∈ G ( F q ) . � θ ( g ) = h ∈ G ( F q ) / B ( F q ) ◮ Note that N G ( B ) = B , and also G ( F q ) / B ( F q ) = ( G / B )( F q ) since by Lang’s theorem H 1 ( F q ; 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 = GL n , 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 − 1 gh ∈ B } ◮ We have two natural maps π : ˜ G ։ G by sending ( g , hB ) to g and ρ : ˜ G ։ T by sending h − 1 gh ∈ B to its image in T . ◮ There is a general way to associate characters on T ( F q ) to sheaves on T . For the moment, let us look at the baby case θ = 1 and we take the constant sheaf Q ℓ T on T . Consider K := π ! ρ ∗ Q ℓ T . ◮ Claim. We have f K = ˜ θ as functions on G ( F q ).

Springer theory ˜ G := { ( g , hB ) ∈ G × G / B | h − 1 gh ∈ B } . ◮ Now the highlight is: ◮ Proposition. We have K [dim G ] := π ! ρ ∗ Q ℓ T [dim G ] ∈ Perv( G ). It is semisimple, with components in bijection with components of the regular representation of W . Here W = S n when G = GL n . ◮ Proof. The only highlight is that ˜ G → G is small . Once we have this, π ! ρ ∗ Q ℓ T is determined by π on a sufficiently small Zariski open of G , which one might take to be G rs the open subvariety of regular semisimple elements; for G = GL n this is those with distinct eigenvalues. Then π − 1 G rs → G rs is finite ´ etale with Galois group W (= S n if G = GL n ). Hence we have π ´ et 1 ( G rs ) → W and π ! Q ℓπ − 1 G rs factors through the regular representation of W . Since K (up to shift) is the middle extension of π ! Q ℓπ − 1 G rs , the result follows. ◮ In fact, for each ρ ∈ Irr( W ) (irreducible Q ℓ -representations of W ) let F ρ ∈ Perv( G ) be the corresponding simple object, then we have a Q ℓ [ W ]-equivariant isomorphism: K [dim G ] ∼ � F ρ ⊗ ρ = ρ ∈ Irr( W ) where the latter ρ is treated as a Q ℓ [ W ]-module.

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 , B ′ , B ′′ ) ∈ G × G / B × G / B | g ∈ B ′ ∩ B ′′ } . G × G ˜ ˜ ◮ 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 ∩ w B ) = dim B − l ( w ). Hence the strata always have dimension = dim G . ◮ All such stratum has dense image in G as B ∩ w B ⊃ 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.

Decomposition of K K [dim G ] := π ! ρ ∗ Q ℓ T [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 ⊗ Q ℓ [ B ( F q )] Q ℓ [ G ( F q )]. ◮ We also have θ ) ∼ End Q ℓ [ G ( F q )] (˜ w ∈ B ( F q ) \ G ( F q ) / B ( F q ) Hom B ( F q ) ∩ w B ( F q ) ( θ, w θ ) as � = 1) ∼ vector spaces. In particular End Q ℓ [ G ( F q )] (˜ = Q ℓ [ W ] as vector spaces. ◮ The last isomorphism can be proved to be an algebra isomorphism. ◮ It is thus natural to expect irreducible components of End Q ℓ [ G ( F q )] (˜ 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 = GL n . ◮ Sketch. When G = GL n , W = S n . Class functions of S n can be generated by inductions from trivial representation of S m 1 × ... × S m k ⊂ S n with m 1 + ... + m k = n . Such a subgroup corresponds to P ⊂ GL n the group of blockwise upper triangular matrices with blocks of size m i . And the induction corresponds to a K P given by replacing B by P in the construction of K .

Another point of view g = { ( X , hB ) ∈ g × G / B | h − 1 Xh ∈ Lie B } . ◮ Let g = Lie G , and ˜ Denote by π g := ˜ g → g the natural map. ◮ Consider N ⊂ g to be the closed subvariety of nilpotent elements, and ˜ N := π − 1 g ( N ). Write π N : ˜ N → N . We also have ◮ Proposition π g is small and π N is semi-small. ◮ The proof is the same; there are again strata indexed by W of the g and ˜ N × N ˜ same dimension in ˜ g × g ˜ 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 = GL n , and u = Lie U . We have ˜ N = { ( X , hB ) | h − 1 Xh ∈ u } , and each stratum of N × N ˜ ˜ N has image being conjugates of u ∩ w u . ◮ Example G = GL 3 . Then the stratum indexed by id ∈ S 3 has image being conjugates of u , namely the whole N . The stratum indexed by (321) ∈ S 3 has image in u ∩ w u = 0. All the rest has images in conjugates of 0 � u ∩ w u � u , which is N sub ⊂ N consisting of nilpotent elements whose Jordan blocks have size ≤ 2.

Springer theory II ◮ What we saw is that in the case G = GL 3 , the map π N : ˜ N → N has π N Q ℓ ˜ N [dim N ] ∈ Perv( N ) and (thanks to decomposition theorem) its image has some components supported on 0 and on N sub . ◮ And it very much looks like, though unclear how, that these components have to do with W . ◮ Write i : N ֒ → g , K g := π g Q ℓ ˜ g [dim G ] and N [dim N ]. We have i ∗ [dim N − dim G ] K g = K N by K N = ( π N ) ∗ Q ℓ ˜ proper base change. Recall that W acts on K g and thus induces a W -action on K N . ◮ Theorem. (Borho-MacPherson) Let i : N ֒ → g . The functor i ∗ [dim N − dim G ] takes simple sub-objects of K G to simple sub-objects of K N . ◮ In particular, each simple sub-object of K N ∈ 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.