On Higher Multiplicity upon Restriction from $\mathrm{GL}(n)$ to $\mathrm{GL}(n-1)$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912141181714432 |
|---|---|
| author | Qadri, Mohammed Saad |
| author_facet | Qadri, Mohammed Saad |
| contents | Let $F$ be a non-archimedean local field. Let $Π$ be a principal series representation of $\mathrm{GL}_n(F)$ induced from an irreducible cuspidal representation of a Levi subgroup. When $π$ is an essentially square integrable representation of $\mathrm{GL}_{n-1}(F)$ we prove that $\mathrm{Hom}_{\mathrm{GL}_{n-1}}(Π,π) = \mathbb{C}$ and $\mathrm{Ext}^i_{\mathrm{GL}_{n-1}}(Π,π) = 0$ for all integers $i\geq 1$, with exactly one exception (up to twists), namely, when $Π= ν^{-(\frac{n-1}{2})} \times ν^{-(\frac{n-3}{2})} \times \ldots \times ν^{(\frac{n-1}{2})}$ and $π$ is the Steinberg. When $Π= ν^{-(\frac{n-1}{2})} \times ν^{-(\frac{n-3}{2})} \times \ldots \times ν^{(\frac{n-1}{2})}$ and $π$ is the Steinberg of $\mathrm{GL}_{n-1}(F)$, then $\dim \mathrm{Hom}_{\mathrm{GL}_{n-1}(F)}(Π,π)=n$. 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 $\mathrm{GL}_{n-1}(F)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_06698 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Higher Multiplicity upon Restriction from $\mathrm{GL}(n)$ to $\mathrm{GL}(n-1)$ Qadri, Mohammed Saad Representation Theory 11F70(primary), 22E55(secondary) Let $F$ be a non-archimedean local field. Let $Π$ be a principal series representation of $\mathrm{GL}_n(F)$ induced from an irreducible cuspidal representation of a Levi subgroup. When $π$ is an essentially square integrable representation of $\mathrm{GL}_{n-1}(F)$ we prove that $\mathrm{Hom}_{\mathrm{GL}_{n-1}}(Π,π) = \mathbb{C}$ and $\mathrm{Ext}^i_{\mathrm{GL}_{n-1}}(Π,π) = 0$ for all integers $i\geq 1$, with exactly one exception (up to twists), namely, when $Π= ν^{-(\frac{n-1}{2})} \times ν^{-(\frac{n-3}{2})} \times \ldots \times ν^{(\frac{n-1}{2})}$ and $π$ is the Steinberg. When $Π= ν^{-(\frac{n-1}{2})} \times ν^{-(\frac{n-3}{2})} \times \ldots \times ν^{(\frac{n-1}{2})}$ and $π$ is the Steinberg of $\mathrm{GL}_{n-1}(F)$, then $\dim \mathrm{Hom}_{\mathrm{GL}_{n-1}(F)}(Π,π)=n$. 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 $\mathrm{GL}_{n-1}(F)$. |
| title | On Higher Multiplicity upon Restriction from $\mathrm{GL}(n)$ to $\mathrm{GL}(n-1)$ |
| topic | Representation Theory 11F70(primary), 22E55(secondary) |
| url | https://arxiv.org/abs/2308.06698 |