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 defining ingredient::centralizer ring for $$B_n(F)$$ in $$GL_n(F)$$ with respect to $$R$$, where $$B_n(F)$$ is a defining ingredient::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 defining ingredient::Iwahori-Hecke algebra of the symmetric group of degree $$n$$.