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Euler characteristics of the real points of certain varieties of algebraic tori
Author(s) -
Lehrer G. I.,
van Hamel J.
Publication year - 2007
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdm002
Subject(s) - mathematics , euler characteristic , pure mathematics , torus , euler's formula , sign (mathematics) , lie algebra , algebraic group , nilpotent , variety (cybernetics) , reductive group , algebra over a field , algebraic number , mathematical analysis , geometry , group theory , statistics
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.