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


We consider the space of $Endo\_K(\tau)$-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?



Twisted Gelfand pairs (Reference and examples)
These Hecke algebras are intensively studied in the field of "type theory" for reductive $p$-adic groups.


You have a nice summary of basic facts *with proofs* in chapter 4 of Bushnell and Kutzko's book "The admissible dual of ${\rm GL}(N)$ via compact open subgroups" (the chapter is entitled "Interlude with Hecke algebras"). 


You may also read the monography "The Langlands conjecture for ${\rm GL}(2)$", written by Bushnell and Henniart. You'll find there a nice introduction to these algebras.


There are many other references. But it depends on what exactly you're interested in.


