arXiv subjects

Shubhanshi Gupta

Publications and source records attributed to Shubhanshi Gupta.

2 recordsLinked to original sources

arXiv2025

On the restriction of some irreducible mod $p$ representations

For a prime $p,$ let $\mathbb{F}_q$ be a finite extension of $\mathbb{F}_p.$ The restriction of an irreducible mod $p$ representation of $\text{GL}_2(\mathbb{F}_q)$ to its subgroup $\text{GL}_2(\mathbb{F}_p)$ can be seen as a tensor product of irreducible representations of $\text{GL}_2(\mathbb{F}_p).$ In this paper, we study the restriction of some of these representations of $\text{GL}_2(\mathbb{F}_q)$ to $\text{GL}_2(\mathbb{F}_p),$ for $q=p^2$ and $p^3$ using elementary tools and give explicit socle filtration when $q=4.$ We prove that when $q=p^2,$ a special class of representations of $\text{GL}_2(\mathbb{F}_q)$ are distinguished by suitable characters of $\text{GL}_2(\mathbb{F}_p).$

math.RT↗

arXiv2025

Restriction problem for mod $p$ representations of $\text{GL}_2$ over a finite field

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.

math.RT↗