Character sheaves and almost characters of reductive groups. I, II.

*(English)*Zbl 0832.20065Let \(G\) be a connected, reductive algebraic group defined over \({\mathbf F}_q\), \(F : G \to G\) a Frobenius morphism and \(G^F\) the finite reductive group of \(F\)-fixed points of \(G\). The famous 1976 paper of P. Deligne and G. Lusztig [Ann. Math., II. Ser. 103, No. 1, 103-161 (1976; Zbl 0336.20029)] was a spectacular breakthrough in the theory of characteristic-zero representations of finite reductive groups. In that paper families of virtual characters of \(G^F\) over \(\overline {\mathbf Q}_l\) (where \(l\) is a prime not dividing \(q\)) parametrized by characters of subgroups of \(G^F\) of the form \(T^F\), where \(T\) is an \(F\)-stable maximal torus of \(G\), were constructed using \(l\)-adic cohomology. Thus we have the Deligne-Lusztig map \(R^G_T : {\mathcal R}(T^F) \to {\mathcal R}(G^F)\), where \({\mathcal R}(G^F)\) denotes the group of virtual characters of \(G^F\). Lusztig defined a more general map \(R^G_L : {\mathcal R} (L^F) \to {\mathcal R} (G^F)\), where \(L\) is an \(F\)-stable Levi subgroup of \(G\). In a series of papers and in his book [Characters of reductive groups over a finite field (Ann. Math. Stud. 107, 1984; Zbl 0556.20033)] G. Lusztig classified all the irreducible characters of \(G^F\), provided \(G\) has a connected center. (This restriction was removed by him later.) This classification leads to a new orthonormal basis of the space \({\mathcal C}(G^F)\) of \(\overline {\mathbf Q}_l\)-valued class functions of \(G^F\) known as the basis of “almost characters”. Lusztig gave a method involving a “Fourier transform” (which works case-by-case) to obtain the irreducible characters from the almost characters. If \(G^F = \text{GL} (n,q)\) or \(\text{U} (n,q)\) the almost characters are (up to sign) the irreducible characters.

The classification of the characters \(G^F\) also led Lusztig to develop a geometric theory of character sheaves on reductive groups over an algebraically closed field. Character sheaves are certain perverse sheaves in the bounded derived category \({\mathcal D} G\) of constructible \(\overline {\mathbf Q}_l\)-sheaves on \(G\). If \(L\) is a Levi-subgroup of \(G\) then one can define induction of character sheaves from \(L\) to \(G\), analogous to the \(R^G_L\) map for class functions on \(L^F\). In our case when the field is \(\overline {\mathbf F}_q\), let \(p\) be the characteristic. One can construct, for an \(F\)-stable complex \(K \in {\mathcal D}G\) with a given isomorphism \(\varphi : F^* K \overset \sim {} K\), its characteristic function \(\chi_{K,\varphi}\), which is a function on \(G^F\). When \(p\) is “almost good” (which is no restriction on classical groups, and a mild restriction on exceptional groups) Lusztig gave a classification of the character sheaves on \(G\), analogous to the classification of the irreducible characters of \(G^F\), and showed that the characteristic functions \(\chi_A\) (\(= \chi_{A,\varphi}\) for suitable \(\varphi)\), where \(A\) runs over the \(F\)-stable character sheaves of \(G\), also form an orthonormal basis of \({\mathcal C} (G^F)\). The computation of the \(\chi_A\) is reduced to the determination of certain “generalized Green functions” on the set of unipotent elements of \(G^F\), and Lusztig gave an algorithm to determine these functions. Furthermore, he conjectured that the functions \(\chi_A\) coincide, up to a scalar multiple, with the almost characters. He showed [J. Am. Math. Soc. 5, No. 4, 971-986 (1992; Zbl 0773.20011)] that his conjecture holds for large \(p\) in the cases where the \(R^G_L\) map is explicitly known, e.g. when \(G\) has a connected center. Thus the problem of determining the character tables of the groups \(G^F\) depends on proving Lusztig’s conjecture in general.

In these two papers the author proves Lusztig’s conjecture for groups \(G\) with a connected center, when \(p\) is almost good and \(q\) is arbitrary. In the first paper (referred to as I) the result is proved for \(G\) of type \(A_n\), \(B_n\), \(C_n\), \(G_2\) or \(F_4\). In the second paper (referred to as II) the author proves the result for types \(E_6\), \(E_7\), \(E_8\) and also gives a simplified and unified proof for classical groups.

The author starts with a result of Lusztig (I, Theorem 1.11) which states that if \(p\) is almost good and \(q\) is sufficiently large then the Lusztig map \(R^G_L\) and the induction map of character sheaves coincide (up to sign). Thus the characteristic function of a complex \(K \in {\mathcal D} G\) which is obtained by inducing an \(F\)-stable “cuspidal” character sheaf \(\varepsilon\) on an \(F\)-stable Levi subgroup \(L\) can be written, up to sign, as \(R^G_L(\chi_\varepsilon)\), where \(\chi_\varepsilon\) is the characteristic function of \(\varepsilon\). Let now \(G\) be a classical group with connected center. The author shows (II, 3.4.2) that Lusztig’s result implies that most of the almost characters can be written as characteristic functions of character sheaves, up to scalar multiples. One of the main steps then is to prove the same result for all \(q\), which is done in (II, 3.7). For this the author uses the method of lifting theory or Shintani descent, which compares functions on \(G^{F^m}\) with functions on \(G^F\) (for suitable \(m\)). A new and interesting feature here is a “Shintani descent identity” for character sheaves (II, Proposition 2.11); the analogue of this for characters was studied by the author, Asai, Digne and Michel, and Lusztig. Finally the case of the remaining character sheaves, which are cuspidal, is worked out; the fact that \(G\) has a connected center plays a role here.

In the case of exceptional groups, the main tool is a twisting operator \(t^*_1\) on \({\mathcal C} (G^F)\) which arises from Shintani descent from \(G^F\) to \(G^F\). The fact that almost characters are eigenfunctions of \(t^*_1\) was known by previous work of Asai and Lusztig. The author proves (I, Theorem 3.3) that the functions \(\chi_A\) arising from character sheaves are also eigenfunctions of \(t^*_1\). These facts are used to prove Lusztig’s conjecture.

Finally the author proves some results on character sheaves when \(p\) is not almost good.

These two papers give the final steps in the proof of Lusztig’s conjecture, and thus pave the way towards implementation of the computation of the character tables of finite reductive groups. (The scalar multiple involved in the conjecture remains to be determined and the author makes a contribution to this problem in a recent paper [“On the computation of unipotent characters of finite classical groups”, in Proc. Conf. on Finite Reductive Groups/Luminy, 1994 (to appear)].) One of the aims of the CHEVIE system developed at Aachen and Heidelberg is to implement Lusztig’s algorithms to construct “generic character tables” of these groups. We have indeed come a long way from the character tables of \(\text{PSL} (2, p)\) and \(\text{SL}(2,q)\) given by Frobenius and Schur respectively in 1896 and 1907.

The classification of the characters \(G^F\) also led Lusztig to develop a geometric theory of character sheaves on reductive groups over an algebraically closed field. Character sheaves are certain perverse sheaves in the bounded derived category \({\mathcal D} G\) of constructible \(\overline {\mathbf Q}_l\)-sheaves on \(G\). If \(L\) is a Levi-subgroup of \(G\) then one can define induction of character sheaves from \(L\) to \(G\), analogous to the \(R^G_L\) map for class functions on \(L^F\). In our case when the field is \(\overline {\mathbf F}_q\), let \(p\) be the characteristic. One can construct, for an \(F\)-stable complex \(K \in {\mathcal D}G\) with a given isomorphism \(\varphi : F^* K \overset \sim {} K\), its characteristic function \(\chi_{K,\varphi}\), which is a function on \(G^F\). When \(p\) is “almost good” (which is no restriction on classical groups, and a mild restriction on exceptional groups) Lusztig gave a classification of the character sheaves on \(G\), analogous to the classification of the irreducible characters of \(G^F\), and showed that the characteristic functions \(\chi_A\) (\(= \chi_{A,\varphi}\) for suitable \(\varphi)\), where \(A\) runs over the \(F\)-stable character sheaves of \(G\), also form an orthonormal basis of \({\mathcal C} (G^F)\). The computation of the \(\chi_A\) is reduced to the determination of certain “generalized Green functions” on the set of unipotent elements of \(G^F\), and Lusztig gave an algorithm to determine these functions. Furthermore, he conjectured that the functions \(\chi_A\) coincide, up to a scalar multiple, with the almost characters. He showed [J. Am. Math. Soc. 5, No. 4, 971-986 (1992; Zbl 0773.20011)] that his conjecture holds for large \(p\) in the cases where the \(R^G_L\) map is explicitly known, e.g. when \(G\) has a connected center. Thus the problem of determining the character tables of the groups \(G^F\) depends on proving Lusztig’s conjecture in general.

In these two papers the author proves Lusztig’s conjecture for groups \(G\) with a connected center, when \(p\) is almost good and \(q\) is arbitrary. In the first paper (referred to as I) the result is proved for \(G\) of type \(A_n\), \(B_n\), \(C_n\), \(G_2\) or \(F_4\). In the second paper (referred to as II) the author proves the result for types \(E_6\), \(E_7\), \(E_8\) and also gives a simplified and unified proof for classical groups.

The author starts with a result of Lusztig (I, Theorem 1.11) which states that if \(p\) is almost good and \(q\) is sufficiently large then the Lusztig map \(R^G_L\) and the induction map of character sheaves coincide (up to sign). Thus the characteristic function of a complex \(K \in {\mathcal D} G\) which is obtained by inducing an \(F\)-stable “cuspidal” character sheaf \(\varepsilon\) on an \(F\)-stable Levi subgroup \(L\) can be written, up to sign, as \(R^G_L(\chi_\varepsilon)\), where \(\chi_\varepsilon\) is the characteristic function of \(\varepsilon\). Let now \(G\) be a classical group with connected center. The author shows (II, 3.4.2) that Lusztig’s result implies that most of the almost characters can be written as characteristic functions of character sheaves, up to scalar multiples. One of the main steps then is to prove the same result for all \(q\), which is done in (II, 3.7). For this the author uses the method of lifting theory or Shintani descent, which compares functions on \(G^{F^m}\) with functions on \(G^F\) (for suitable \(m\)). A new and interesting feature here is a “Shintani descent identity” for character sheaves (II, Proposition 2.11); the analogue of this for characters was studied by the author, Asai, Digne and Michel, and Lusztig. Finally the case of the remaining character sheaves, which are cuspidal, is worked out; the fact that \(G\) has a connected center plays a role here.

In the case of exceptional groups, the main tool is a twisting operator \(t^*_1\) on \({\mathcal C} (G^F)\) which arises from Shintani descent from \(G^F\) to \(G^F\). The fact that almost characters are eigenfunctions of \(t^*_1\) was known by previous work of Asai and Lusztig. The author proves (I, Theorem 3.3) that the functions \(\chi_A\) arising from character sheaves are also eigenfunctions of \(t^*_1\). These facts are used to prove Lusztig’s conjecture.

Finally the author proves some results on character sheaves when \(p\) is not almost good.

These two papers give the final steps in the proof of Lusztig’s conjecture, and thus pave the way towards implementation of the computation of the character tables of finite reductive groups. (The scalar multiple involved in the conjecture remains to be determined and the author makes a contribution to this problem in a recent paper [“On the computation of unipotent characters of finite classical groups”, in Proc. Conf. on Finite Reductive Groups/Luminy, 1994 (to appear)].) One of the aims of the CHEVIE system developed at Aachen and Heidelberg is to implement Lusztig’s algorithms to construct “generic character tables” of these groups. We have indeed come a long way from the character tables of \(\text{PSL} (2, p)\) and \(\text{SL}(2,q)\) given by Frobenius and Schur respectively in 1896 and 1907.

Reviewer: B.Srinivasan (MR 95k:20069)

##### MSC:

20G05 | Representation theory for linear algebraic groups |

20C40 | Computational methods (representations of groups) (MSC2010) |

20G40 | Linear algebraic groups over finite fields |

20C33 | Representations of finite groups of Lie type |