Green functions and character sheaves.

*(English)*Zbl 0695.20024Let 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 \(Q^ G_ T\).

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 |