Hecke algebra of the general linear group

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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.