Characters of finite reductive Lie algebras

For any finite group of Lie type G(q), Deligne and Lusztig [P. Deligne, G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976) 103-161] defined a family of virtual (Q) over bart-characters R-T(G)(theta) of G(q) such that any irreducible T character of G(q) is an irreducible constituent of at least one of the R-T(G)(theta). In this paper we study analogues of this result for char- acters of the finite reductive Lie algebra G(q) where G = Lie(G). Motivated by the results of [E. Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Lecture Notes in Math., vol. 1859. Springer-Veriag, 2005] and (G. Luszrig, Representations of reductive groups over finite rings, Represent. Theory 8 (2004) 1-14], we define two families R-T(G) and R-T(G) (theta) of virtual T T (Q) over barl-characters of G(q). We prove that they coincide when theta is in general position and that they differ in general. We verify that ally character of G(q) appears in some R-T(G) (theta). We conjecture that this T C is also true if R-T(G) is replaced by R-T(G). (C) 2009 Elsevier Inc. All rights reserved.