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Algebra Seminar
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Gus Lehrer
University of Sydney

Euler characteristics of varieties of real algebraic tori

Friday 20th August, 12:05-12:55pm, Carslaw 175.

Let G be a complex connected reductive Lie group which is defined over R, let g be its Lie algebra, and T the variety of maximal tori of G. For x in g(R), let Tx be the variety of tori in T whose Lie algebra is orthogonal to x with respect to the Killing form. This is a complex algebraic variety which is defined over R. I shall explain how the Fourier-Sato transform of conical sheaves on real vector bundles may be used to show that the "weighted Euler characteristic" of Tx(R) is zero unless x is nilpotent, in which case it equals (-1)½dimT. This Euler characteristic therefore provides a formula for the characteristic function of the real nilpotent cone. Alternatively it could be thought of as providing a remarkable characterisation of nilpotent elements in real Lie algebras.

This and other similar results are analogues of results concerning the Steinberg character of a finite reductive group and its Lie algebraic analogue.