BC_n-symmetric polynomials

Abstract

We consider two important families of BC_n-symmetric polynomials, namely Okounkov's interpolation polynomials and Koornwinder's orthogonal polynomials. We give a family of difference equations satisfied by the former, as well as generalizations of the branching rule and Pieri identity, leading to a number of multivariate q-analogues of classical hypergeometric transformations. For the latter, we give new proofs of Macdonald's conjectures, as well as new identities, including an inverse binomial formula and several branching rule and connection coefficient identities. We also derive families of ordinary symmetric functions that reduce to the interpolation and Koornwinder polynomials upon appropriate specialization. As an application, we consider a number of new integral conjectures associated to classical symmetric spaces.

65 pages, LaTeX. v2-3: Minor corrections and additions (including teasers for the sequel). v4: C^+ notation changed to harmonize with the sequels (and more teasers added)

References in corpus (5)

• Difference equations and symmetric polynomials defined by their zeros
• Symmetric and non-symmetric quantum Capelli polynomials
• Self-dual Koornwinder-Macdonald polynomials
• On certain multiple Bailey, Rogers and Dougall type summation formulas
• A proof of a multivariable elliptic summation formula conjectured by Warnaar

Cited by in corpus (2)

• Macdonald-level extension of beta ensembles and large-N limit transition
• Laguerre and Meixner orthogonal bases in the algebra of symmetric functions