I have an algebraic reductive group $G$ (say $GL(n)$) over a number field $F$ (say $\mathbf{Q}_p$), and $K$ a maximal compact subgroup of $G(F)$ (say $GL(n, \mathcal{O}_p)$. Fix a left Haar measure on $G$.
Take a finite union of $K$-double cosets $T = \bigcup_A K \alpha K$. For an element $\beta \in G$, I am interested in computing volumes of the form $$\mathrm{vol} \left( T \cap \beta K \right).$$
However, $T$ also decomposes as a union of left cosets $\bigcup_C \gamma K$, and hence the volume should be either $0$ or $\mathrm{vol}(K)$ according to whether or not $\beta K$ is among the $\gamma K$.
Is that true or am I missing something?
Because this fact gives me results I really would like to avoid.