Let $p$ be a prime number, $q=p^e$ a power of $p$, and $G=SL_2(\mathbb F_q)$. Let $V$ be the adjoint representation of $G$, i.e. $V$ is the 3-dimensional $\mathbb F_q$-space of of (2,2)-matrices of trace $0$ with coefficients in $\mathbb F_q$, and $G$ acts on it by conjugation. So $V$ is an irreducible representation of $G$.

By "restriction of scalar", $V$ can also be considered as a representation of dimension $3 e$ over $\mathbb F_p$.

When is this representation $V/\mathbb F_p$ of dimension $3e$ irreducible? absolutely irreducible?

In general, I would be interested by any pointer to a reference mentioning questions of irreducibility of representations obtained by "restriction of scalars".

Let $p$ be a prime number, $q=p^e$ a power of $p$, and $G=SL_2(\mathbb F_q)$. Let $V$ be the adjoint representation of $G$, i.e. $V$ is the 3-dimensional $\mathbb F_q$-space of of (2,2)-matrices of trace $0$ with coefficients in $\mathbb F_q$, and $G$ acts on it by conjugation. So $V$ is an irreducible representation of $G$.

By "restriction of scalar", $V$ can also be considered as a representation of dimension $3 e$ over $\mathbb F_p$.

When is this representation $V/\mathbb F_p$ of dimension $3e$ irreducible?

(Edit: it is obviously not absolutely irreducible, as remarked by Venkataramana below).

In general, I would be interested by any pointer to a reference mentioning questions of irreducibility of representations obtained by "restriction of scalars".

Let $p$ be a prime number, $q=p^e$ a power of $p$, and $G=SL_2(\mathbb F_q)$. Let $V$ be the adjoint representation of $G$, i.e. $V$ is the 3-dimensional $\mathbb F_q$-space of of (2,2)-matrices of trace $0$ with coefficients in $\mathbb F_q$, and $G$ acts on it by conjugation. So $V$ is an irreducible representation of $G$.

By "restriction of scalar", $V$ can also be considered as a representation of dimension $3 e$ over $\mathbb F_p$.

When is this representation $V/\mathbb F_p$ of dimension $3e$ irreducible? absolutely irreducible"?

In general, I would be interested to any pointer mentioning question of irreducibility for representations obtained by "restriction of scalars".

Let $p$ be a prime number, $q=p^e$ a power of $p$, and $G=SL_2(\mathbb F_q)$. Let $V$ be the adjoint representation of $G$, i.e. $V$ is the 3-dimensional $\mathbb F_q$-space of of (2,2)-matrices of trace $0$ with coefficients in $\mathbb F_q$, and $G$ acts on it by conjugation. So $V$ is an irreducible representation of $G$.

By "restriction of scalar", $V$ can also be considered as a representation of dimension $3 e$ over $\mathbb F_p$.

When is this representation $V/\mathbb F_p$ of dimension $3e$ irreducible? absolutely irreducible?

In general, I would be interested by any pointer to a reference mentioning questions of irreducibility of representations obtained by "restriction of scalars".