The answer to the question below is almost certainly known to the representation theorists; in fact, I'm pretty sure it can be extracted from Green's paper "The characters of the finite general linear groups" in Trans. Amer. Math. Soc. 80 (1955), 402–447. However, I am not a specialist in representation theory (I work in geometry), so I am having trouble extracting it from the literature.

This preamble brings me to my question. Fix some prime number $p$. Consider $1 \leq m \leq n$. Let $\vec{e}_1,\ldots,\vec{e}_n \in \mathbb{F}_p^n$ be the standard basis and let $\Gamma_{n,m}$ be the subgroup of $\text{GL}_n(\mathbb{F}_p)$ that stabilizes the vectors $\vec{e}_1,\ldots,\vec{e}_m$ point wise. Define $V_{n,m}$ to be the $\text{GL}_n(\mathbb{F}_p)$-representation obtained by inducing the $1$-dimensional trivial representation $\mathbb{C}$ of $\Gamma_{n,m}$ up to $\text{GL}_n(\mathbb{F}_p)$.

**Question** : What is the decomposition of $V_{n,m}$ into irreducible representations? Or at least how many terms appear in this decomposition (as a function of $n$ and $m$)? I have some reason (hope?) that the answer to this only depends on $m$ once $n$ is sufficiently large.

Finally, can anyone recommend an easy-to-read source for reading about the sort of representation theory that would go into answering this kind of question? Preferably one that is as concrete as possible (so one that e.g. doesn't assume that I am an expert in reductive groups).

**EDIT** : Some of the confusion in the comments comes from the fact that I changed the way I stated the question after I started writing it to be a little more "slick"; this made the question not obviously correspond to the title.

Here's the original formulation (which involved parabolic induction more directly). Define $\overline{\Gamma}_{n,m}$ to be the $\text{GL}_n(\mathbb{F}_p)$-stabilizer of the flag $$0 < \langle \vec{e}_1,\ldots,\vec{e}_m \rangle < \mathbb{F}_p^n.$$ Thus $\Gamma_{n,m} < \overline{\Gamma}_{n,m}$ and $\overline{\Gamma}_{n,m}$ is the parabolic subgroup which is the semidirect product of $\text{GL}_m(\mathbb{F}_p) \times \text{GL}_{n-m}(\mathbb{F}_p)$ and a unipotent subgroup $U_{n,m}$. If we induce the trivial $\Gamma_{n,m}$-representation $\mathbb{C}$ up to $\overline{\Gamma}_{n,m}$, we get a representation $\overline{V}_{n,m}$ of $\overline{\Gamma}_{n,m}$ which is the $\mathbb{C}$-group ring of $\text{GL}_{m}(\mathbb{F}_p)$ on which $\text{GL}_{n-m}(\mathbb{F}_p)$ and $U_{n,m}$ act trivially. The irreducible subrepresentations of $\overline{V}_{n,m}$ are thus exactly the irreducible representations of $\text{GL}_{m}(\mathbb{F}_p)$. This brings us to the following, which is similar to the question above (the first part of it would give an answer to the first question above).

**Question** : Let $M$ be an irreducible representation of $\text{GL}_{m}(\mathbb{F}_p)$. Regard $M$ as a representation of $\overline{\Gamma}_{n,m}$, and let $\tilde{M}$ be the result of inducing $M$ from $\overline{\Gamma}_{n,m}$ up to $\text{GL}_n(\mathbb{F}_p)$. How can we decompose $\tilde{M}$ into irreducibles? Can we at least get a bound on the number of irreducibles in $\tilde{M}$ as a function of $n$ and $m$ (which even better only depends on $m$ as long as $n$ is sufficiently large)?