Ring structure of Hecke algebra For any locally profinite group $G$ and its open compact subgroup $K$，one can define the hecke algebra $H(G,K)$ as the space of compact supported bi $K$-invariant functions on $G$ with convolution as product operation.
Hecke algebra is very important in the representation theory of $G$, and there are several results about commutativity of such algebra. While we can always write a basis for such algebra using characteristic function for double coset, I know little about it's ring structure. For example, $H(GL_n(\Bbb Q_p),GL_n(\Bbb Z_p))$ is commutative, but I wonder what it is as a ring.
Therefore, are there some results about the ring structure? Are there some nice examples and applications of the explicit ring structure?
 A: As is typical in Lie theory, if you want to find an explicit calculation, it's best to start by checking if Macdonald wrote something about it. In this case, you should start by having a look at Spherical functions on a $\mathfrak{p}$-adic Chevalley group, available free here
https://projecteuclid.org/download/pdf_1/euclid.bams/1183529627
This short paper gives an explicit isomorphism (Theorem 1') from the spherical Hecke algebra $L(G,U)$ of compactly supported, continuous, $U$-bi-invariant functions from $G$ to $\mathbf{C}$ to $\mathbf{C}[P^\vee]^W$, where $G$ is a Chevalley group, $U$ is a maximal compact subgroup, $P^\vee$ is the coweight lattice, $W$ is the Weyl group, so by Bourbaki (Chapter 6 of Lie groups and Lie algebras), $\mathbf{C}[P^\vee]^W$ is a polynomial ring on $r=\mathrm{rk}(G)$ generators.
So even though it doesn't directly address your question (only since $\mathrm{GL}_n$ is reductive but not quite semisimple), you will find it easier to read than his book Symmetric functions and Hall polynomials, which contains the relevant material for the general linear group (in which case the spherical Hecke algebra is the algebra of symmetric polynomials).
