Appears in collection : Symmetry in Geometry and Analysis

Let $G$ be a real reductive algebraic group, and let $H$ be an algebraic subgroup of $G$. 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 of Schwartz functions on $G/H$ are finite in this case. I will talk about a recent joint work with A. Aizenbud in which we formulate and analyze a generalization of sphericity that implies finite multiplicities in the Schwartz space of $G/H$ for small enough irreducible smooth representations of $G$.

I will also report on another joint work, with E. Sayag, in which we give a geometric sufficient condition for vanishing of multiplicities. In more detail, for every $G$-space $X$, and every closed $G$-invariant subset $S$ of the nilpotent cone of the Lie algebra of $G$, we define when $X$ is $S$-spherical, by means of a geometric condition involving dimensions of fibers of the moment map. We then show that if $X$ is $S$-spherical, then every representation with annihilator variety lying in $S$ has (at most) finite multiplicities in the Schwartz space of $X$. We give applications of our results to branching problems. Our main tool in bounding the multiplicity is the theory of holonomic $\mathcal D$-modules. After formulating our main results, I will briefly recall the necessary aspects of this theory and sketch our proofs.