### 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
*T*_{x} 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 *T*_{x}(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.