Super-Macdonald Polynomials: Orthogonality and Hilbert Space Interpretation

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