On Higher Multiplicity upon Restriction from $\mathrm{GL}(n)$ to $\mathrm{GL}(n-1)$

Abstract

Let $F$ be a non-archimedean local field. Let $\hat{I}$ be a principal series representation of ${\mathrm{GL}}_n(F)$ induced from an irreducible cuspidal representation of a Levi subgroup. When $\ddot{I}€$ is an essentially square integrable representation of ${\mathrm{GL}}_{n-1}(F)$ we prove that ${\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}}(\hat{I},\ddot{I}€)=\mathbb{C}$ and ${\mathrm{Ext}}_{{\mathrm{GL}}_{n-1}}^i(\hat{I},\ddot{I}€)=0$ for all integers $i\ge 1$, with exactly one exception (up to twists), namely, when $\hat{I}=\hat{I}{½}^{-(\frac{n-1}{2})}\times \hat{I}{½}^{-(\frac{n-3}{2})}\times \dots \times \hat{I}{½}^{(\frac{n-1}{2})}$ and $\ddot{I}€$ is the Steinberg. When $\hat{I}=\hat{I}{½}^{-(\frac{n-1}{2})}\times \hat{I}{½}^{-(\frac{n-3}{2})}\times \dots \times \hat{I}{½}^{(\frac{n-1}{2})}$ and $\ddot{I}€$ is the Steinberg of ${\mathrm{GL}}_{n-1}(F)$, then $\dim {\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}(F)}(\hat{I},\ddot{I}€)=n$. We also exhibit specific principal series for which each of the intermediate multiplicities $2,3,\cdots ,(n-1)$ are attained. Along the way, we also give a complete list of those irreducible non-generic representations of ${\mathrm{GL}}_n(F)$ that have the Steinberg of ${\mathrm{GL}}_{n-1}(F)$ as a quotient upon restriction to ${\mathrm{GL}}_{n-1}(F)$. We also show that there do not exist non-generic irreducible representations of ${\mathrm{GL}}_n(F)$ that have the generalized Steinberg as a quotient upon restriction to ${\mathrm{GL}}_{n-1}(F)$.