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I am trying to understand a particular coset/double coset of the finite group $G = GL(n, q^2) = GL_n(\mathbb{F}_{q^2})$. It has a natural subgroup $H = GL(n, q)$, which can also be viewed in the following way: consider an automorphism of raising each entry to the $q$-th power, (taking $n = 2$ as an example)

$$ \varphi\begin{bmatrix}a & b \\ c & d\end{bmatrix} = \begin{bmatrix}a^q & b^q \\ c^q & d^q\end{bmatrix}, $$

clearly $\varphi^2 = Id$, and $H = G^{\varphi}$ as the fixed points of this morphism.

Question: what is a good way to view the coset $G/H$ and double coset $H \backslash G/H$, and any good way of writing the representatives?

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In general, let $K \to L$ be a field extension. $GL_n(L)/GL_n(K)$ can be interpreted as the set of "$K$-structures" on $L^n$. One of many equivalent ways to describe a $K$-structure is that it is a $K$-subspace $V$ of $L^n$ such that the induced map

$$V \otimes_K L \to L^n$$

is an isomorphism. Consequently, $GL_n(K) \backslash GL_n(L) / GL_n(K)$ can be described as the set of "relative positions" of two $K$-structures, in the sense of e.g. this blog post.

Edit, 7/28/16: In turn you can get some handle on relative positions by thinking about functions or properties of a pair $V, W$ of $K$-structures that are invariant under ($L$-linear) change of coordinates. A simple example is $\dim_K V \cap W$. More generally, for any $\ell \in L^{\times}$, you can look at $\dim_K \ell V \cap W$.

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A standard heuristic to get a feeling for such a question in my world is to replace $\mathbb F_{q^2}$ by $\mathbb F_q \oplus \mathbb F_q$, so that the question is about $GL(n,\mathbb F_q)\times GL(n,\mathbb F_q)$ modulo a diagonal copy of $GL(n,\mathbb F_q)$, and about the corresponding double cosets. The first question is very easy, then. The second yields the answer that the double cosets are in bijection with conjugacy classes in $GL(n,\mathbb F_q)$.

After gathering one's optimism, then one goes back and tries to see what happens in the original... thinking that it's a "Galois twist" of the "split case".

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