In this talk I will describe a very interesting interaction between geometry
and representation theory which produces new results in both subjects.
I will try to make the presentation accessible to people with different
background, only some familiarity with representations of compact groups
will be assumed.
Recently W. Schmid and K. Vilonen proved two character formulas for certain
representations of real reductive Lie groups $G$
– the fixed point character formula and the integral character formula.
In the case when $G$ is compact, the former reduces to the
Weyl character formula and the latter – to Kirillov’s character formula.
These representations were constructed by M.~Kashiwara and W.~Schmid.
They generalize the Borel-Weil-Bott construction, but instead of line
bundles on the flag variety they consider $G$-equivariant constructible
sheaves ${\cal F}$ and, for each integer $p$, they define representations of
$G$ in $\operatorname{Ext}^p({\cal F},{\cal O})$,
where ${\cal O}$ is the sheaf of functions on the flag variety.
In this talk I will explain these two character formulas and outline my
geometric proof of equivalence of these two formulas.
The corresponding problem for compact groups was solved by
N.~Berline and M.~Vergne using their famous integral localization formula
for equivariant cohomology. This new geometric argument leads to
a generalization of their localization formula to non-compact group actions.
Based on math.RT/0208024.