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Proceedings of the London Mathematical Society Advance Access originally published online on March 21, 2007
Proceedings of the London Mathematical Society 2007 94(3):715-748; doi:10.1112/plms/pdm002
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© 2007 London Mathematical Society

Euler characteristics of the real points of certain varieties of algebraic tori

G. I. Lehrer1 and J. van Hamel2

1 School of Mathematics and Statistics
The University of Sydney
Sydney
NSW 2006
Australia
gusl{at}maths.usyd.edu.au
2 Department of Mathematics
KU Leuven
Celestijnenlaan 200B
B-3001 Leuven (Heverlee)
Belgium
Joost.vanHamel{at}wis.kuleuven.be

Received 8 March 2005. Revision received 29 March 2006.

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} isin G(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.


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