Multiplicity One Theorems: The Archimedean Case

Let $G$ be one of the classical Lie groups $GL_n(\Bbb R), GL_n(\Bbb C)$, $O(p,q)$, $O_n(\Bbb C) U(p,q)$, and let $G’$ be respectively the subgroup $GL_{n-1}(\Bbb R)$, $GL_{n-1}(\Bbb C), O(p,q-1), O_{n-1}(\Bbb C), U(p, q-1)$, embedded in $G$ in the standard way. We show that every irreducible Harish-Chandra smooth representation of $G’$ occurs with multiplicity at most one in every irreducible Harish-Chadnra smooth representation of $G$. This is joint work with Binyong Sun of the Chinese Academy of Sciences.

Independently and in a different approach, A.Aizenbud and D.Gourevitch have proved the multiplicty one theorems for the pairs $(GL_n(\Bbb R), GL_{n-1}(\Bbb R))$ and $(GL_n(\Bbb C), GL_{n-1}(\Bbb C))$.