Centralizer ring

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.