A NONSYMMETRIC VERSION OF OKOUNKOV’S BC-TYPE INTERPOLATION MACDONALD POLYNOMIALS

Nonsymmetric interpolation Laurent polynomials in n variables are introduced, with the interpolation points depending on q and on a n-tuple of parameters τ = (τ1, …, τn). When τi = stn − 1, Okounkov’s 3-parameter BCn-type interpolation Macdonald polynomials are recovered from the nonsymmetric interpolation Laurent polynomials through Hecke algebra symmetrisation with respect to a type Cn Hecke algebra action. In the Appendix we give some conjectures about extra vanishing, based on Mathematica computations in rank two.

Elliptic Ruijsenaars Difference Operators on Bounded Partitions

By means of a truncation condition on the parameters, the elliptic Ruijsenaars difference operators are restricted onto a finite lattice of points encoded by bounded partitions. A corresponding orthogonal basis of joint eigenfunctions is constructed in terms of polynomials on the joint spectrum. In the trigonometric limit, this recovers the diagonalization of the truncated Macdonald difference operators by a finite-dimensional basis of Macdonald polynomials.

Super-Macdonald Polynomials: Orthogonality and Hilbert Space Interpretation

The super-Macdonald polynomials, introduced by Sergeev and Veselov (Commun Math Phys 288: 653–675, 2009), generalise the Macdonald polynomials to (arbitrary numbers of) two kinds of variables, and they are eigenfunctions of the deformed Macdonald–Ruijsenaars operators introduced by the same authors in Sergeev and Veselov (Commun Math Phys 245: 249–278, 2004). We introduce a Hermitian form on the algebra spanned by the super-Macdonald polynomials, prove their orthogonality, compute their (quadratic) norms explicitly, and establish a corresponding Hilbert space interpretation of the super-Macdonald polynomials and deformed Macdonald–Ruijsenaars operators. This allows for a quantum mechanical interpretation of the models defined by the deformed Macdonald–Ruijsenaars operators. Motivated by recent results in the nonrelativistic ($$q\rightarrow 1$$ q → 1 ) case, we propose that these models describe the particles and anti-particles of an underlying relativistic quantum field theory, thus providing a natural generalisation of the trigonometric Ruijsenaars model.