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Green functions and character sheaves. (English) Zbl 0695.20024
Let G be a connected, reductive algebraic group defined over $$F_ q$$, and let $$F: G\to G$$ be a Frobenius morphism. Let $$G^ F$$ be the finite group of F-fixed points of G. Corresponding to each F-stable maximal torus T, P. Deligne and the author [Ann. Math., II. Ser. 103, 103-161 (1976; Zbl 0336.20029)] defined functions $$Q^ G_ T$$, known as Green functions, on the set of unipotent elements of $$G^ F$$. Later the author gave a new definition of Green functions using intersection cohomology and in his work on character sheaves [Adv. Math. 61, 103-155 (1986; Zbl 0602.20036)] gave a method of computing these functions in principle. The coincidence of the two definitions of Green functions was known for large p (where q is a power of p) by work of Springer and Kazhdan, and for all “good” p in some cases by further work of the author [J. Algebra 104, 146-194 (1986; Zbl 0603.20037)]. In this paper the author proves that the two definitions coincide for all p, provided q is sufficiently large. More generally, he considers the generalized Green functions defined by him via intersection cohomology [in Adv. Math. 57, 226-265 (1985; Zbl 0586.20019)] and connects them with functions obtained via twisted induction from a Levi subgroup of G, when p is “almost good”.
Let $${\mathcal D}G$$ denote the bounded derived category of constructible $$\bar Q_{\ell}$$-sheaves on G (here $$\ell$$ is a prime distinct from p). The abelian category $${\mathcal M}G$$ of perverse sheaves on G is a full subcategory of $${\mathcal D}G$$, and character sheaves are certain objects in $${\mathcal M}G$$. In analogy with the Harish-Chandra theory for $$G^ F$$, certain character sheaves are called cuspidal. If K is a complex in $${\mathcal D}G$$ with a given isomorphism $$\phi$$ : $$F^*K\overset \sim \rightarrow K$$, then one gets a $$\bar Q_{\ell}$$-valued characteristic function $$\chi_{K,G}$$ on $$G^ F$$ by setting $$\chi_{K,\phi}(x)=\sum_{i\geq 0}(-1)^ iTr(\phi,H^ i_ xK)$$ where $$H^ i_ xK$$ is the stalk at x of the i-th cohomology sheaf $$H^ iK$$ of K.
A cuspidal character sheaf on a Levi subgroup M of G gives rise to an induced complex in $${\mathcal D}G$$ which is a direct sum of character sheaves on G. More precisely, suppose we have the following data. Let M be an F- stable Levi subgroup of a parabolic subgroup MV of G with unipotent radical V. Let $$\Sigma$$ be the inverse image under $$M\to M/Z^ 0_ M$$ (here $$Z_ M$$ is the center of M) of a single, F-stable conjugacy class of $$M/Z^ 0_ M$$. Let $${\mathcal E}$$ be an M-equivariant (for the conjugation action of M) $$\bar Q_{\ell}$$-local system on $$\Sigma$$ which gives rise, by extension first to the closure $${\bar \Sigma}$$ of $$\Sigma$$ as an intersection cohomology complex and then to M by 0 on $$M-{\bar \Sigma}$$, to a complex $${\mathcal E}^{\#}\in {\mathcal D}M$$ such that $${\mathcal E}^{\#}[\dim \Sigma]$$ is a direct sum of cuspidal character sheaves on M. (Here [ ] denotes shift). Let $$\tau$$ : $$F^*{\mathcal E}\overset \sim \rightarrow {\mathcal E}$$ be an isomorphism. Then, to this data is associated an induced complex $$K\in {\mathcal D}G$$ and an isomorphism $${\bar \tau}$$: $$F^*K\overset \sim \rightarrow K$$. Now assume further that $$\Sigma =CZ^ 0_ M$$ where C is an F-stable unipotent conjugacy class of M, $${\mathcal E}={\mathcal F}\otimes \bar Q_{\ell}$$ where $${\mathcal F}$$ is an M-equivariant $$\bar Q_{\ell}$$-local system on C, and that $$\tau =\tau_ 0\otimes 1$$ where $$\tau_ 0: F^*{\mathcal F}\overset \sim \rightarrow {\mathcal F}$$ is an isomorphism. Then the restriction of the characteristic function $$\chi_{K,{\bar \tau}}$$ to the set of unipotent elements of $$G^ F$$ is called a generalized Green function and is denoted by $$Q^ G_{M,C,{\mathcal F},\tau_ 0}$$. In particular, taking $$G=M$$, $$K={\mathcal E}^{\#}[\dim \Sigma]$$ we have $$Q^ M_{M,C,{\mathcal F},\tau_ 0}=(-1)^{\dim \Sigma}\chi_{{\mathcal E}^{\#},\tau}$$. By applying the twisted induction map $$R^ G_{M,V}$$ (which generalizes Deligne-Lusztig induction) to the function $$Q^ M_{M,C,{\mathcal F},\tau_ 0}$$ we get another function $$\bar Q^ G_{M,V,C,{\mathcal F},\tau_ 0}$$ on the set of unipotent elements of $$G^ F$$. The main theorem of this paper (1.14) is then as follows. There is a constant $$q_ 0>1$$ depending only on the Dynkin diagram of G, such that if $$q\geq q_ 0$$ the following hold:
(a) If p is “almost good” for M, then $$(-1)^{\dim \Sigma}\tilde Q^ G_{M,V,C,{\mathcal F},\tau_ 0}=Q^ G_{M,C,{\mathcal F},\tau_ 0}.$$
(b) Suppose p is “almost good” for G. Then the space $$F^ G_ G$$ spanned by functions on $$G^ F$$ of the form $$Q^ G_{G,C,{\mathcal F},\tau_ 0}$$ (i.e. restrictions to the unipotent elements of the characteristic functions of cuspidal character sheaves of G, defined over $$F_ q)$$ is precisely the space of $$\bar Q_{\ell}$$-valued functions on the unipotent elements of $$G^ F$$ which are orthogonal to the functions of the form $$R^ G_{M,V}(f)$$ for any $$M\neq G$$ and any class function $$f: M^ F\to \bar Q_{\ell}$$. In the case when M is a maximal torus T we can take $${\mathcal F}=\bar Q_{\ell}$$, $$C=\{e\}$$, and $$\tau =1.$$
In this case (Proposition 8.15) the assumptions of the theorem can be weakened and the conclusion is that the generalized Green function $$\tilde Q^ G_ T$$ is equal to the Deligne-Lusztig Green function $$Q^ G_ T$$.
Reviewer: B.Srinivasan

##### MSC:
 20G05 Representation theory for linear algebraic groups 20C15 Ordinary representations and characters 14L30 Group actions on varieties or schemes (quotients) 20G10 Cohomology theory for linear algebraic groups 14F30 $$p$$-adic cohomology, crystalline cohomology 14L40 Other algebraic groups (geometric aspects) 20G40 Linear algebraic groups over finite fields
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