Spherical subgroups of Kac–Moody groups and transitive actions on spherical varieties

On Orbit Closures of Borel Subgroups in Spherical Varieties

Let F be the flag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on F with finitely many orbits, and let V be an H-orbit closure in F . Expanding the cohomology class of V in the basis of Schubert classes defines a union V0 of Schubert varieties in F with positive multiplicities. If G is simply-laced, we show that these multiplicites are equal to the same p...

On Orbit Closures of Spherical Subgroups in Flag Varieties

Let F be the ag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on F with nitely many orbits, and let V be an H-orbit closure in F. Expanding the cohomology class of V in the basis of Schubert classes deenes a union V 0 of Schubert varieties in F with positive multiplicities. If G is simply-laced, we show that these multiplicites are equal to the same power ...

Intersection Theory on Spherical Varieties

Convergence of spherical averages for actions of free groups

Let (X, ν) be a probability space and suppose a free group Fm withm generators acts on (X, ν) by measure-preserving transformations. Let {a1, . . . , am} be a set of free generators for Fm and let T1, . . . , Tm : X → X be transformations corresponding to the generators. Write T−i = T −1 i for i = 1, . . . ,m, and set A = {−m, . . . ,−1, 1, . . . ,m}. We also have the action Fm on L1(X, ν), def...

Toric Degenerations of Spherical Varieties

We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by Mirror Symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose exist...