Toshiaki Shoji

Length functions and Demazure operators for G(e,1,n), II

Abstract This note is the first part of consecutive two papers concerning with a length function and Demazure operators for the complex reflection group W = G(e, 1, n). In this first part, we study the word problem on W based on the work of Bremke and Malle [BM]. We show that the usual length function l(W) associated to a given generator set S is completely described by the function n(W), introduced in [BM], associated to the root system of W. In the second part, we will study the Demazure operators of W on the symmetric algebra. We define a graded space HW in terms of Demazure operators, and show that HW is isomorphic to the coinvariant algebra SW, which enables us to define a homogeneous basis on SW parametrized by w ϵ W.

Algebraic Groups and Quantum Groups

Let g be a complex simple Lie algebra, f a nilpotent element of g. We show that (1) the center of the W -algebra Wcri(g, f) associated (g, f) at the critical level coincides with the Feigin-Frenkel center of ĝ, (2) the centerless quotient Wχ(g, f) of Wcri(g, f) corresponding to an Lg-oper χ on the disc is simple, and (3) the simple quotient Wχ(g, f) is a quantization of the jet scheme of the intersection of the Slodowy slice at f with the nilpotent cone of g.

A norm map for endomorphism algebras of Gelfand-Graev representations

Let G be a connected, reductive algebraic group defined over a finite field, with Frobenius endomorphism F: G → G. In a previous article by the first author [3], the irreducible representations, in an algebraically closed field K of characteristic zero, of the Hecke algebra Ή of a Gelfand-Graev representation γ of the finite reductive group G F were determined. They are parametrized by pairs (T, θ), with T an F-stable maximal torus of G, and θ an irreducible character of the finite torus T F . Each irreducible representation f T, θ can be factored, \({{f}_{{T,{\kern 1pt} \theta }}}{\mkern 1mu} = {\mkern 1mu} \tilde{\theta } {\mkern 1mu} ^\circ {{f}_{T}} \), with f T a homomorphism of algebras, independent of θ, from H to the group algebra KTF, and \({\tilde{\theta }} \) the extension of θ to an irreducible representation of the group algebra of the torus T F .

Unipotent characters of finite classical groups

Let G be a connected reductive group defined over a finite field F q with Frobenius map F. One of the main problems in the representation theory of finite groups is to determine all the irreducible characters (on C or \({{\overline {\text{Q}} }_{l}} \)) of G F and complete their character tables. Lusztig classified all the irreducible characters of G F and determined their degrees ([L2]). But as far as the character values are concerned, it is still open. In order to attack this problem, Lusztig proposed a general strategy for it and gave a conjecture. It is explained roughly as follows. Let ν be the \({{\overline {\text{Q}} }_{l}} \)-vector space of class functions of G F . As is well-known the set of irreducible characters of G F forms an orthonormal basis of ν under the natural inner product. He constructed two other types of orthonormal basis of ν, i.e., the one consisting of almost characters of G F , and the other consisting of characteristic functions of F-stable character sheaves on G.

Green functions associated to complex reflection groups, II

Green functions associated to complex reflection groups G(e,1,n) were discussed in the authors previous paper. In this paper, we consider the case of complex reflection groups W=G(e,p,n). Schur functions and Hall–Littlewood functions associated to W are introduced, and Green functions are described as the transition matrix between those two symmetric functions. Furthermore, it is shown that these Green functions are determined by means of Green functions associated to various G(e′,1,n′). Our result involves, as a special case, a combinatorial approach to the Green functions of type Dn.

Generalized Green functions and unipotent classes for finite reductive groups, II

Lusztigs algorithm of computing generalized Green functions of reductive groups involves an ambiguity of certain scalars. In this paper, for reductive groups of classical type with arbitrary characteristic, we determine those scalars explicitly, and eliminate the ambiguity. Our results imply that all the generalized Green functions of classical type are computable.

Lusztig’s conjecture for finite special linear groups

In this paper, we prove Lusztigs conjecture for finite special linear groups, i.e., we show that characteristic functions of character sheaves coincide with almost characters up to scalar constants, under the condition that the characteristic of the base field is not too small. We determine those scalars explicitly. Our result gives a method of computing irreducible characters of these groups.

Subfield symmetric spaces for finite special linear groups

Let

Quadratic equations over finite fields and class numbers of real quadratic fields

G

On the computation of unipotent characters of finite classical groups

be a finite reductive group defined over a finite field