# Hecke algebra of the general linear group

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## Definition

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

If is a finite field with elements, the Hecke algebra of can be obtained by specializing to the value in the Iwahori-Hecke algebra of the symmetric group of degree .