Hecke algebra of the general linear group
From Groupprops
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
.