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 -characters of G(q) such that any irreducible character of G(q) is an irreducible constituent of at least one of the . 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 and of virtual -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 . We conjecture that this is also true if is replaced by .