An algebraic approach to Macdonald–Koornwinder polynomials: Rodrigues-type formula and inner product identity

Abstract

We study Macdonald–Koornwinder polynomials in the context of double affine Hecke algebras. Nonsymmetric Macdonald–Koornwinder polynomials are constructed by use of raising operators provided by a representation theory of the double affine Hecke algebra associated with A2l(2)-type affine root system. This enables us to evaluate diagonal terms of scalar products of the nonsymmetric polynomials algebraically. The Macdonald–Koornwinder polynomials are expressed by linear combinations of the nonsymmetric counterparts. We show a new proof of the inner product identity of the Macdonald–Koornwinder polynomials without Opdam–Cherednik’s shift operators.