Elliptic Analogues of the Macdonald and Koornwinder Polynomials

摘要

Perhaps the nicest multivariate orthogonal polynomials are the Macdonald and Koornwinder polynomials, respectively 2-parameter deformations of Schur functions and 6-parameter deformations of orthogonal and symplectic characters, satisfying a trio of nice properties known as the Macdonald "conjectures". In recent work, the author has constructed elliptic analogues: a family of multivariate functions on an elliptic curve satisfying analogues of the Macdonald conjectures, and degenerating to Macdonald and Koornwinder polynomials under suitable limits. This article will discuss the two main constructions for these functions, focusing on the more algebraic/combinatorial of the two approaches.