# Centralizer ring

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## Definition

### Definition with symbols

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

• It is the endomorphism ring of the $R[G]$-module $R[G/B \times G/B]$ where the module action is defined by coordinate-wise left multiplication by $G$
• It is the endomorphism ring of the $R[B]$-module $R[G/B]$ where $B$ acts on the coset space by left multiplication.
• It is the endomorphism ring of the $R$-module over the double coset space of $B$ in $G$.

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