Hecke algebra of an algebraic group

Definition

Let $G$ be an algebraic group over a field $k$ and $B$ be a Borel subgroup of $G$. The Hecke algebra of $G$ (corresponding to $B$) over a commutative unital ring $R$ is defined as the centralizer ring for $B$ in $G$ with respect to $R$.

Note that $R$ need not be equal to the field $k$.