Equivariant sheaves on some spherical varieties

In this announcement we describe categories of equivariant vector bundles on certain “spherical” varieties. Our description is in linear-algebra terms: vector spaces equipped with filtrations, group and Lie-algebra actions, and linear maps preserving these structures. The two parents of our description are i) the description of G-equivariant vector bundles on homogeneous spaces G/H as H-representations and ii) Klyachko’s description of the category of equivariant vector bundles on a toric variety (cf. [Kly89]) in terms of vector spaces equipped with families of filtrations satisfying a compatibility condition. Results in this spirit were first presented by Kato (cf. [Kat05]), and we recover and extend his results in several directions. Detailed proofs of our results will appear in [AP]. We begin with an illustrative example, whose description prefigures the general situation. Let k be a field of characteristic 0. Working in the category of k-schemes, let X = P × P equipped with the diagonal (left) action of G = PGL2. Let T be the stabilizer in G of the point x with bihomogeneous coordinates ([1 : 1], [0 : 1]). Let Vec(X) denote the category of G-equivariant vector bundles on X. Let Filt(k) be the category whose objects are triples (V, ρ, F ), where V is a finite-dimensional k-vector space, where ρ is a representation of T on V , and where F • is a decreasing finite filtration on V . Morphisms in Filt(k) are T-module homomorphisms compatible with the filtrations.