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Euler characteristics of the real points of certain varieties of algebraic tori. (English) Zbl 1122.14040

Authors’ abstract: Let \(G\) be a complex connected reductive group which is defined over \(\mathbb{R}\), let \(\mathfrak{G}\) be its Lie algebra, and let \(\mathcal{T}\) be the variety of maximal tori of \(G\). For \(\xi \in \mathfrak{G} (\mathbb{R})\), let \(\mathcal{T}_\xi\) be the variety of tori in \(\mathcal{T}\) 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 \(\mathcal{T}_\xi (\mathbb{R})\) is zero unless \(\xi\) is nilpotent, in which case it equals \((-1)^{\frac{\dim\mathcal{T}}{2}}\). Here ‘weighted Euler characteristic’ means the sum of the Euler characteristics of the connected components, each weighted by a sign \(\pm 1\) 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.

MSC:

14P25 Topology of real algebraic varieties
22E15 General properties and structure of real Lie groups
57T15 Homology and cohomology of homogeneous spaces of Lie groups
20G40 Linear algebraic groups over finite fields
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