On the character values of finite Chevalley groups at unipotent elements
Let \(G(F_ q)\) be the group of rational points of a reductive connected algebraic group G over a finite field \(F_ q\). The author presents a method for computing the values of the irreducible characters of \(G(F_ q)\) at unipotent elements. The author's method can be explained as follows. The space of class functions on \(G(F_ q)\) has two natural orthonormal bases: one, \((f_ i)\), consists of the irreducible characters, the other one, \((f_ j')\) consists of the characteristic functions of certain irreducible perverse sheaves on G, called character sheaves. Then to compute the \((f_ i)\) on the unipotent elements it is enough to solve two other problems: (a) compute the \(f_ j'\) at unipotent elements. (b) compute the transition matrix between the bases \((f_ i)\), \((f_ j').\) Problem (a) is closely related to the problem of computing the local intersection cohomology of the closure of a unipotent class of G with coefficients in a G-equivariant irreducible local system. Problem (b) is more difficult. There is a known formula which expresses those \(f_ i\) which are components of a principal series representation in terms of the \(f_ j'\). This information is quite powerful for solving problem (b). The author asserts that his method can be used to determine the Green functions of \(G(F_ q)\) assuming that G is adjoint simple, split over \(F_ q\) and q is subject to the following conditions: (*) Type A: no condition; type B: q odd; type C, \(D: q\equiv 1 (mod 4)\); type \(G_ 2\), \(E_ 6:\) \(q\equiv 1 (mod 6)\); type \(F_ 4\), \(E_ 7:\) \(q\equiv 1 (mod 12)\); type \(E_ 8:\) \(q\equiv 1 (mod 60).\) The author also asserts that he can use his method to compute for G as above, the character at unipotent elements of any irreducible representation of \(G(F_ q)\) such that the corresponding semisimple element in the dual group \(G^*(F_ q)\) has centralizer of maximal semisimple rank. These include, for example, all unipotent representations of \(G(F_ q)\). The author gives explicit formulae for the values of an arbitrary irreducible character of \(G(F_ q)\) at unipotent elements, when G is adjoint, split of type \(B_ n\) or \(E_ n\) (q as in (*)).
scientific article; zbMATH DE number 3973161
scientific article