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

Abstract: Let $F$ be a non-archimedean local field. Let $\Pi$ be a principal series representation of $\mathrm{GL}n(F)$ induced from an irreducible cuspidal representation of a Levi subgroup. When $\pi$ is an essentially square integrable representation of $\mathrm{GL}{n-1}(F)$ we prove that $\mathrm{Hom}{\mathrm{GL}{n-1}}(\Pi,\pi) = \mathbb{C}$ and $\mathrm{Ext}^i_{\mathrm{GL}_{n-1}}(\Pi,\pi) = 0$ for all integers $i\geq 1$, with exactly one exception (up to twists), namely, when $\Pi= \nu^{-(\frac{n-1}{2})} \times \nu^{-(\frac{n-3}{2})} \times \ldots \times \nu^{(\frac{n-1}{2})}$ and $\pi$ is the Steinberg. When $\Pi= \nu^{-(\frac{n-1}{2})} \times \nu^{-(\frac{n-3}{2})} \times \ldots \times \nu^{(\frac{n-1}{2})}$ and $\pi$ is the Steinberg of $\mathrm{GL}{n-1}(F)$, then $\dim \mathrm{Hom}{\mathrm{GL}{n-1}(F)}(\Pi,\pi)=n$. 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 $\mathrm{GL}{n-1}(F)$.