Restriction problem for mod $p$ representations of $\text{GL}_2$ over a finite field
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| Format: | Preprint |
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2025
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| _version_ | 1866909651366313984 |
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| author | Ghate, Eknath Gupta, Shubhanshi |
| author_facet | Ghate, Eknath Gupta, Shubhanshi |
| contents | Let $\mathbb{F}_q$ be the finite field with $q = p^f$ elements. We study the restriction of two classes of mod $p$ representations of $G_q = \text{GL}_2({\mathbb{F}_q})$ to $G_p = \text{GL}_2(\mathbb{F}_p)$. We first study the restrictions of principal series which are obtained by induction from a Borel subgroup $B_q$. We then analyze the restrictions of inductions from an anisotropic torus $T_q$ which are related to cuspidal representations. Complete decompositions are given in both cases according to the parity of $f$. The proofs depend on writing down explicit orbit decompositions of $G_p \backslash G_q / H$ where $H = B_q$ or $T_q$ using the fact that $G_q / H$ is an explicit orbit in a certain projective line, along with Mackey theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_14207 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Restriction problem for mod $p$ representations of $\text{GL}_2$ over a finite field Ghate, Eknath Gupta, Shubhanshi Representation Theory Number Theory Let $\mathbb{F}_q$ be the finite field with $q = p^f$ elements. We study the restriction of two classes of mod $p$ representations of $G_q = \text{GL}_2({\mathbb{F}_q})$ to $G_p = \text{GL}_2(\mathbb{F}_p)$. We first study the restrictions of principal series which are obtained by induction from a Borel subgroup $B_q$. We then analyze the restrictions of inductions from an anisotropic torus $T_q$ which are related to cuspidal representations. Complete decompositions are given in both cases according to the parity of $f$. The proofs depend on writing down explicit orbit decompositions of $G_p \backslash G_q / H$ where $H = B_q$ or $T_q$ using the fact that $G_q / H$ is an explicit orbit in a certain projective line, along with Mackey theory. |
| title | Restriction problem for mod $p$ representations of $\text{GL}_2$ over a finite field |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/2506.14207 |