On the restriction of some irreducible mod-$p$ representations of $\text{GL}_2(\mathbb{F}_q)$ to $\text{GL}_2(\mathbb{F}_p)$

Abstract: For a prime $p,$ let $\mathbb{F}_q$ be a finite extension of $\mathbb{F}_p$ and $\mathcal{G}=\text{GL}_2(\mathbb{F}_q).$ Then the irreducible representations of $\mathcal{G}$ are classified as twists of Sym$^{\vec{r}}(\overline{\mathbb F}_p^2).$ The restriction of irreducibles of $\mathcal{G}$ to its subgroup $G=\text{GL}_2(\mathbb{F}_p)$ is same as investigating the behavior of the tensor product of irreducible representations of $G.$ In this paper, we study the restriction of some of these representations of $\mathcal{G}$ to $G,$ for $2$ and $3$ degree extensions of $\mathbb{F}_p.$