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Remark: I cross-posted this question on MSE and added a bounty to it. Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly ...

$\DeclareMathOperator\GL{GL}$Let $F$ be a $\mathfrak{p}$-adic field and $\mathscr{O}_{F}$ its valuation ring. For any measurable subset of $M_{d}(F)$ such as $$ A= \left( \begin{array}{ccc} a_{11}+t^{\...

Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as $$...

Let $(G,K)$ be a Gelfand pair. Why, for a function $f$ $K$-binvariant with respect to a compact subgroup $K$ of a group $G$, do we have the following equality: $$ f(xy) = \int_K f(xky) \, dk$$ A ...

Let $\alpha$ be a continuous automorphism on a compact group $G$ with normalized Haar measure $m$. We may say $\alpha$ is $n$-splitting, if the set $$\text{Spl}_n(\alpha):=\left\{g\in G: \prod_{k=1}^...