Let $GL_n(F_q)$ be the general linear group over finite field $F_q$ and $B_n$ be its borel subgroup consisting of all upper triangular matrices. Then the double cosets $B_n\backslash GL_n(F_q)/B_n$ are parametrised by the permutation group $S_n$, which can be viewed as permutation matrices. Further by the theory of representations of symmetric groups, it follows that the number of symmetric permutation matrices is equal to the number of irreducible representations(with multiplicities) of the Hecke algebra $C[B_n\backslash C[B_n\backslash GL_n(F_q)/B_n]$.

My question is: how far is it true? In the sense that suppose I am given a general matrix group (with entries not necessarily from a field) $G$ with a subgroup $B$ such that double cosets $B\backslash G/B$ can be parametrised by symmetric matrices, then is it true that the Hecke algebra $C[B\backslash C[B\backslash G/B]$ is commutative?

Let $GL_n(F_q)$ be the general linear group over finite field $F_q$ and $B_n$ be its borel subgroup consisting of all upper triangular matrices. Then the double cosets $B_n\backslash GL_n(F_q)/B_n$ are parametrised by the permutation group $S_n$, which can be viewed as permutation matrices. Further by the theory of representations of symmetric groups, it follows that the number of symmetric permutation matrices is equal to the number of irreducible representations(with multiplicities) of the Hecke algebra $C[B_n\backslash GL_n(F_q)/B_n]$.

My question is: how far is it true? In the sense that suppose I am given a general matrix group $G$ with a subgroup $B$ such that double cosets $B\backslash G/B$ can be parametrised by symmetric matrices, then is it true that the Hecke algebra $C[B\backslash G/B]$ is commutative?