Hecke Algebra Solutions to the Reflection Equation

We construct solutions to Sklyanin's reeection equation in the case in which the bulk Yang-Baxter solution is of Hecke algebra type. Each solution constitutes an extension of the Hecke algebra together with a spectral parameter dependent boundary operator, K(). We solve for the deening relations of the extension, and for the spectral parameter dependence of the boundary operator. Finding concrete matrix solutions to the reeection equation is thus reduced to the problem of constructing nite dimensional representations of the Hecke algebra extension. This problem is solved in the generic case by establishing isomorphism with an aane Hecke algebra quotient whose representation theory is known.