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 $\vec{e}_1,\ldots,\vec{e}_k$ 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 $m$)? 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).