Finite multiplicity theorem for spherical pairs

Let $(G, H)$ be a spherical pair over a local field $k$ and let $\pi$ be an admissible representation of $G$. Kobayashi and Oshima, and independently Kroetz and Schlichtkrull have recently shown that, if $k=\mathbb R$, then the space of $H$-invariant functionals on $\pi$ is finite-dimensional. Both approaches use Casselman's theorem which says that $\pi$ can be presented as a quotient of a principal series representation. We will consider another approach that does not use this theorem. An important step in our proof is to show that the singular support of any $H$-spherical character is a Lagrangian in the cotangent bundle of $G$. In the future, we hope to generalize our proof to the $p$-adic case, where Casselman's theorem does not hold, and the finite multiplicity theorem is known only for certain kinds of spherical pairs (due to Delorme and to Sakellaridis–Venkatesh). The talk is based on an ongoing work with A. Aizenbud and D. Gourevitch.