Finite Multiplicities Beyond Spherical Spaces

Let G be a real reductive algebraic group, and let H ⊂ G be an algebraic subgroup. It is known that the action of G on the space of functions on G/H is “tame” if this space is spherical. In particular, the multiplicities of the space S(G/H) of Schwartz functions on G/H are finite in this case. In this paper, we formulate and analyze a generalization of sphericity that implies finite multiplicities in S(G/H) for small enough irreducible representations of G.