Euler characteristics of the real points of certain varieties of algebraic tori

Abstract. Let G be a complex connected reductive group which is defined over R, let G be its Lie algebra, and T the variety of maximal tori of G. For ξ ∈ G(R), let Tξ be the variety of tori in T whose Lie algebra is orthogonal to ξ with respect to the Killing form. We show, using the Fourier–Sato transform of conical sheaves on real vector bundles, that the “weighted Euler characteristic ” (see below) of Tξ(R) is zero unless ξ is nilpotent, in which case it equals (−1) dimT2. This is a real analogue of a result over finite fields, which is connected with the Steinberg representation of a reductive group. 1. Introduction an