Representation Theory seminar
Wednesday, 13 September 2023, 9:30 am
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Venue: Mini Conference Room
Host: U. K. Anandavardhanan
Speaker: Mohammed Saad Qadri
Affiliation: IIT Bombay
Title: On Higher Multiplicity upon Restriction from GL(n) to GL(n−1)
Abstract:
Let $F$ be a non-archimedean local field. Let $\Pi$ be a principal series representation of $\GL_n(F)$ induced from a cuspidal representation of a Levi subgroup. When $\pi$ is an essentially square integrable representation of $\GL_{n-1}(F)$ we prove that $\Hom_{\GL_{n-1}}(\Pi,\pi)$ $= \mathbb{C}$ and $\Ext^i_{\GL_{n-1}}(\Pi,\pi) = 0$ 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 $\GL_{n-1}(F)$, then $\dim \Hom_{\GL_{n-1}(F)}(\Pi,\pi)=n$. 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 $\GL_{n}(F)$ that have the Steinberg of $\GL_{n-1}(F)$ as a quotient upon restriction to $\GL_{n-1}(F)$. We also show that there do not exist non-generic irreducible representations of $\GL_{n}(F)$ that have the generalized Steinberg as a quotient upon restriction to $\GL_{n-1}(F)$.