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 be its Lie algebra, and let be the variety of maximal tori of G . For ξ ∈ (ℝ), let ξ be the variety of tori in 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’ of ξ (ℝ) is zero unless ξ is nilpotent, in which case it equals (−1) (dim )/2 . Here ‘weighted Euler characteristic’ means the sum of the Euler characteristics of the connected components, each weighted by a sign ± 1 which depends on the real structure of the tori in the relevant component. This is a real analogue of a result over finite fields which is connected with the Steinberg representation of a reductive group.