Hecke algebra of the general linear group

Definition

Suppose $F$ is a field, $R$ is a and $GL_{n}(F)$ is the general linear group of degree $n$ over $F$. Then, the Hecke algebra of $GL_{n}(F)$ over a commutative unital ring $R$ is defined as its Hecke algebra viewing it as an algebraic group. More explicitly, it is the centralizer ring for $B_{n}(F)$ in $GL_{n}(F)$ with respect to $R$, where $B_{n}(F)$ is a Borel subgroup, which can be taken as the group of upper-triangular invertible matrices in $GL_{n}(F)$.

If $F$ is a finite field with $q$ elements, the Hecke algebra of $GL_{n}(F)$ can be obtained by specializing to the value $q$ in the Iwahori-Hecke algebra of the symmetric group of degree $n$.