Notes on Green functions

Notes on Green functions Jean Michel University Paris VII AIM, 4th June, 2007 Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 1 / 15 We consider a reductive group G over the algebraic closure F q of a finite field F q, which is defined over F q. We are interested in the finite group G(F q) = G F, where F is a Frobenius endomorphism corresponding to the F q-structure on G. On A 1 = Spec(F q[x ]), the geometric Frobenius is X X q while the arithmetic Frobenius is i aix i i aq i X i. The composed is the identity on points. For G = GL n, where F raises the matrix coefficients to their q-th power, Green (1955) has defined functions attached to each rational maximal torus, such that the values of characters of G at unipotent elements are given by linear combinations of such functions. We give a survey of these functions for the case of general reductive groups, pointing out along the way which simplifications hold for GL n. Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 2 / 15

Cohomological induction of Deligne and Lusztig G F -classes of maximal tori are parameterized by H 1 (F, W ); given a rational maximal torus T w, Deligne and Lusztig (1976) built a cohomological induction R G which sends characters of T F w to virtual characters of G F. Irreducible characters of G F appear in these virtual characters with small multiplicities independent of q (in the case of GL n, are linear combinations of such characters). R G is defined via l-adic cohomology: the GF -classes of pairs T B such that T is rational are parameterized by W. Given such a pair T w B parameterized by w, let U = R u(b) (so that B = T w U), and let X w = {x G x 1 F (x) U}. We have actions of G F on the left and T F w on the right on this variety, giving rise to actions on the cohomology groups Hc( X i w, Q l ). For θ Irr(T F w ), we define R G (θ)(g) = i ( 1) i Trace(g H i c( X w, Q l ) θ ). Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 3 / 15 Unipotent characters Though X w depends on B, the alternating sum depends only on T w (i.e. on the class of w in H 1 (F, W )). For an element with Jordan decomposition s u, there is a character formula R G (θ)(su) = (#C G(s) 0F ) 1 s) 0F {h G F h s } RCG(h ( h u)θ( h s). which shows that the value is 0 if s does not have a conjugate T F w, and otherwise depends on s only via θ of that conjugate. Thus R G (θ)(u) does not depend on θ. So, when considering values at unipotent elements, it is enough to look at the Unipotent characters, constituents of R G (Id). One has R G (Id), RG T (Id) w G F = δ w,w #C W (wf ), so [to simplify I assume from now on that F acts trivially on W ] one is led to introduce, for χ Irr(W ), the almost characters R χ := (#W ) 1 w W χ(w)rg (Id) which satisfy R χ, R ψ G F = δ χ,ψ. They are the irreducible unipotent characters in the case of GL n. Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 4 / 15

GL 2 (F q ) Classes 0 1 B @ a 0 C A 0 a 0 1 B @ a 0 C A 0 b 0 1 B @ x 0 C A 0 F x 0 1 B @ a 1 C A 0 a # such classes q 1 (q 1)(q 2)/2 q(q 1)/2 q 1 # class 1 q(q + 1) q(q 1) q 2 1 R G (α, β) (q + 1)α(a)β(a) α(a)β(b) + α(b)β(a) 0 α(a)β(a) T1 R G Ts (ω) (1 q)ω(a) 0 ω(x) + ω(f x) ω(a) R G T1 (α, α) (1 + q)α(a2 ) 2α(ab) 0 α(a 2 ) R G Ts.(α N F q 2 /Fq ) (1 q)α(a2 ) 0 2α(x. F x) α(a 2 ) in the above a, b F q, a b and x F q 2, x F x and α, β Irr(F q ), α β, ω Irr(F q 2 ), ω ω q. Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 5 / 15 The variety X w Choosing a Borel subgroup B 0, the variety B of Borel subgroups of G can be identified to G/B 0. If W = N G (T)/T, where T is a maximal torus of B 0, we have the Bruhat decomposition G = w W B0wB0, which implies that the orbits of B B under the diagonal action of G are in bijection with W. We write that B w B if the orbit of (B, B ) is parameterized by w, or equivalently (B, B ) G (B 0, w B 0). Then R G (Id) is given by the cohomology of the simpler variety X w = {B B B w F (B)}: we have R G (Id)(g) = i ( 1)i Trace(g Hc(X i w, Q l )). X w is an étale covering of X w of group T F w, so we could also have defined R G (θ) as the cohomology of a sheaf associated to θ on Xw. By definition the Green function Q w (u) is R G (Id)(u). Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 6 / 15

Parabolic induction and Lusztig induction Let P = U L be a rational Levi decomposition of a parabolic subgroup of G. Then on Q l [G F /U F ] we have a left action of G F and a right action of L F. We define the parabolic induction or Harish-Chandra induction for χ Irr(L F ): R G L (χ)(g) = Trace(g Q l[g F /U F ] χ) = (#L F ) 1 l L F χ(l) Trace((g, l) Q l [G F /U F ]). We have a generalisation of this: if L is still rational but P is not necessarily rational any longer, let X U = {g G g 1 F (g) U}. Then X U admits an action on G F on the left and of L F on the right, and for χ Irr(L F ) we define R G L (χ)(g) = i ( 1) i Trace(g H i c(x U, Q l ) χ). The variety X U depends on P, but the alternating sum depends only on L. Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 7 / 15 The Hecke algebra and the principal series In the case of GL n, all the unipotent characters belong to the principal series, the constituents of the representation R G (Id) Q T1 l[g F /U F 0 ] Id = Q l [G F /B F 0 ] = Ind GF Id B F 0. The commuting algebra End G F (R G ) has a basis given by the operators T1 T w (B) = B B w B, and is the Hecke algebra H(W, q) (a deformation of the algebra of W where the order relations for the Coxeter generators s 2 = 1 are replaced by quadratic relations (T s + 1)(T s q) = 0). It can be shown that H(W, q) Q l [W ], and if one calls χ χ q the corresponding correspondence of characters, one gets the decomposition R G = T1 χ Irr(W ) χ(1)uχ (for GLn we have Rχ = Uχ but not in general), and we can write more generally Trace(T w g R G ) = T1 χq Irr(H(W,q)) χq( )Uχ(g). Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 8 / 15

Lefschetz numbers A way of computing R G is via the Lefschetz trace formula: for any m > 0 we have i ( 1)i Trace(gF m Hc(X i w, Q l )) = #X gf m w. We want the value at m = 0 to do that we may consider the series f (t) = m m=1 #XgF w t m, and we have f ( ) = R G (g). Indeed let V Hc(X i w, Q l ) be a generalized eigenspace of F for the eigenvalue λ and let ρ be the character of G F afforded by V :we have Trace(gF m V ) = ρ(g)λ m, and m=1 tm λ m ρ(g) = ρ(g) λt 1 λt whose limit for t = is indeed ρ(g). Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 9 / 15 Shintani descent Since G is connected, any element of G can be written x 1 F m (x). If x 1 F m (x) G F, then xf (x 1 ) G F m and the correspondence x 1 F m (x) xf (x 1 ) G F m induces a well-defined map N F /F m : H 1 (F, G F ) H 1 (F, G F m ) (H 1 (F, G F ) is just the set of conjugacy classes of G F ). The functions on G F m factoring through H 1 (F, G F m ) are generated by the restrictions to G F m F of characters of G F m F. A basis is obtained by taking the restriction of one extension ρ of each F -invariant character of G F m. We have the Shintani descent identity (Asai, Digne-Michel 1980): #X gf m w where Ũ GF m χ G F m F. = Trace(N F /F m(g)t w F R GF m = χ q m Irr(H(W,q m )) T1 ) χ q m(t w )ŨGF m χ (N F /F m(g)f ), is an extension of the irreducible character U GF m χ Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 10 / 15 to

Green functions and perverse sheaves A difficult result, proved initially by Springer and Kazhdan (1978), then later in a different way by Lusztig and Shoji (1982), is that (for q a large enough power of a large enough prime) Q w (u) = i 0( 1) i Trace(wF H i (B u, Q l )) where B u := {B B B u}. One of the difficulties is to explain the action of W on the cohomology of B u. Lusztig initiated an approach using perverse sheaves. Let X = {(g, B) G B B g}. The proper map X π G : (g, B) x (considered by Grothendieck and Springer, in order to get a resolution of singularities of the unipotent subvariety G uni) is, above the open subset G ss of regular semisimple elements, an étale covering of group W. Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 11 / 15 Local systems Thus L := π Q l, the direct image of the constant sheaf on π 1 (G ss), is a local system on G ss whose fibers are isomorphic to the regular representation of W. Given an irreducible local system L on a locally closed smooth irreducible subvariety Y of a variety X, the intersection cohomology complex IC(Y, L)[dim Y] is a simple perverse sheaf, and any simple perverse sheaf on X is of this form. We will denote L # for IC(Y, L)[dim Y]. The action of W on L gives rise to an action on L #, and Lusztig has shown that L # = π Q l [dim G]. We get thus an action of W on π Q l, and thus on the stalks H i x(π Q l ) = H i (π 1 (x), Q l ) = H i (B x, Q l ). Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 12 / 15

The Springer correspondence Let I be the set of irreducible G-equivariant local systems on G uni; their support is one unipotent class, so an element ι I is given by a pair (E, C) where C is a unipotent class and E is an irreducible G-equivariant local system on C (which amounts to be given a character of A G (u) := C G (u)/cg 0 (u) for u C). Any simple G-equivariant perverse sheaf on G uni is of the form ι #. Borho and MacPherson (1981) described the decomposition in simples as well as the action of W on π Q l Guni ; it is given by: π Q l [dim G uni] = χ Irr(W ) χ ι# χ where χ ι χ : Irr(W ) I is the Springer correspondence. It is surjective for GL n, but not in general (though its image contains trivial local systems on all unipotent classes). For GL n it sends a character defined by some partition to the trivial local system on the unipotent class with the same partition. Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 13 / 15 Characteristic functions Let E be a G-equivariant local system on a locally closed G-stable subvariety of G. Assume that E is F -stable, i.e. there is an isomorphism F (E) φ E. Then one may consider Trace(F E x) for x G F. This defines the characteristic function χ E (or χ φ,e note that, even if E is irreducible, φ is defined only up to scalar, so we get a class function on G F defined up to a scalar). For a perverse sheaf ι # with an isomorphism φ : F ι # ι # one defines similarly χ φ,ι #(x) = i ( 1)i Trace(φ H i (ι # ) x). Let Y ι be the characteristic function of the local system E, if ι = (E, C); for GL n, it is just the charateristic function of C F. We 1 normalize it by setting Ỹ ι = q 2 (dim Guni dim C) Y ι. With the above notations, the values R χ(u) (where, as above, R χ denotes the Mellin transform #W 1 w W χ(w)qw ), are given by R χ(u) = χ φ,ι # = χ ι Pι,ιχỸι(u), where P ι,ι are polynomials in q. Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 14 / 15

The Lusztig-Shoji algorithm (1985) The P ι,ι are determined by the matrix equation: t PΛP = Ω where P is the matrix {P ι,ι } ι,ι. If ι = (E, C) and ι = (E, C ) then P ι,ι = 0 unless C C. And if C = C, we have P ι,ι = δ E,E. Thus, in an appropriate basis, P is unipotent lower triangular. In GL n the relation C C is the dominance order on partitions; if C corresponds to the partition λ 1 λ 2... and C to the partition µ 1 µ 2... the condition is λ 1 +... λ i µ 1 +... µ i for all i. If ι = ι χ and ι = ι χ, then Ω ι,ι = l i=0 ql i ( 1) i χ χ, r i W, where r is the reflection representation of W and l the semi-simple rank of G. Λ ι,ι = Y ι, Y ι G F. Thus, for GL n it is diagonal with diagonal coefficients (#C G F (u)) 1 where u runs over representatives of the unipotent classes. Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 15 / 15