Euler Characteristics of the Real Points of Certain Varieties of Algebraic Tori

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) dimT 2 . This is a real analogue of a result over finite fields, which is connected with the Steinberg representation of a reductive group.