Stable spherical varieties and their moduli

We introduce a notion of stable spherical variety which includes the spherical varieties under a reductive group G and their flat equivariant degenerations. Given any projective space ℙ where G acts linearly, we construct a moduli space for stable spherical varieties over ℙ, that is, pairs (X,f), where X is a stable spherical variety and f:X → ℙ is a finite equivariant morphism. This space is projective, and its irreducible components are rational. It generalizes the moduli space of pairs (X,D), where X is a stable toric variety and D is an effective ample Cartier divisor on X which contains no orbit. The equivariant automorphism group of ℙ acts on our moduli space; the spherical varieties over ℙ and their stable limits form only finitely many orbits. A variant of this moduli space gives another view to the compactifications of quotients of thin Schubert cells constructed by Kapranov and Lafforgue.