Affine spherical varieties and antiholomorphic involutions

It is known that a toric variety possesses a canonical real structure.
A similar structure exists for non-singular affine spherical varieties.
The structure is given by an antiholomorphic involution of the variety, equivariant with respect
to the involutive antiholomorphic automorphism of the reductive transformation group
that defines the split real form. We will sketch the proof of existence of such a real
structure and discuss some applications to geometric representation theory.

Exposé dans les cadres du rencontre
Journées de la géométrie affine à  Grenoble,
Grenoble, 9-10 mai 2011