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Let $G$ be a locally compact group and let $K$ be a compact group. Let $(\tau, V_\tau)$$ V_\tau)$ be an irreducible representation .of $K$.

We consider the space of function $f : G -> \textup{Endo}K (V\tau)$ Endo_K(\tau)$-valued, compactly supported continuous functions $f$ on $G$
with $$ f(k_1 g k_2) = \tau(k_1) f(g) \tau(k_2), $$ which is an $*$ algebra under convolution.

What is a good reference for such algebras, especially in the context with reductive group over local fields and the connection to representation theory?
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Twisted Gelfand pairs (Reference and examples)

Let $G$ be a locally compact group and let $K$ be a compact group. Let $(\tau, V_\tau)$$ be an irreducible representation.

We consider the space of function $f : G -> \textup{Endo}K (V\tau)$ with $$ f(k_1 g k_2) = \tau(k_1) f(g) \tau(k_2), $$ which is an algebra under convolution.

What is a good reference for such algebras, especially in the context with reductive group over local fields?