Proceedings of the London Mathematical Society vol:94 pages:715-748
Let G be a complex connected reductive group which is defined over R, let G be its Lie algebra, and let T be the variety of maximal tori of G. For xi is an element of O(R), let T-xi be the variety of tori in T whose Lie algebra is orthogonal to xi 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 T-xi(R) is zero unless xi is nilpotent, in which case it equals ((-)1) ((dim T) /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.