Characters of finite reductive Lie algebras

Abstract

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 ¯ ℓ -characters R T G ( θ ) of G ( q ) such that any irreducible character of G ( q ) is an irreducible constituent of at least one of the R T G ( θ ) . In this paper we study analogues of this result for characters 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-Verlag, 2005] and [G. Lusztig, Representations of reductive groups over finite rings, Represent. Theory 8 (2004) 1–14], we define two families R T G ( θ ) and R T G ( θ ) of virtual Q ¯ ℓ -characters of G ( q ) . We prove that they coincide when θ is in general position and that they differ in general. We verify that any character of G ( q ) appears in some R T G ( θ ) . We conjecture that this is also true if R T G is replaced by R T G .