# Centralizer ring

## Definition

### Definition with symbols

Given a group , a subgroup , and a ring , the centralizer ring of with respect to the subgroup and over the ring is defined in any of the following equivalent ways:

- It is the endomorphism ring of the -module where the module action is defined by coordinate-wise left multiplication by
- It is the endomorphism ring of the -module where acts on the coset space by left multiplication.
- It is the endomorphism ring of the -module over the double coset space of in .

In the particular case where is an algebraic group and is a Borel subgroup, the corresponding centralizer ring is called the Hecke algebra.