Centralizer ring

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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.