Abstract: | Let G be a complex connected reductive group which is definedover , let be its Lie algebra, and let be the variety of maximaltori of G. For (), let be the variety of tori in whose Liealgebra is orthogonal to with respect to the Killing form.We show, using the FourierSato transform of conical sheaveson real vector bundles, that the weighted Euler characteristicof () is zero unless is nilpotent, in which case it equals(1)(dim )/2. Here weighted Euler characteristicmeans the sum of the Euler characteristics of the connectedcomponents, each weighted by a sign ± 1 which dependson the real structure of the tori in the relevant component.This is a real analogue of a result over finite fields whichis connected with the Steinberg representation of a reductivegroup. |