Orthogonality of super-Jack polynomials and a Hilbert space interpretation of deformed Calogero–Moser–Sutherland operators

Journal article, 2019

We prove orthogonality and compute explicitly the (quadratic) norms for super-Jack polynomials SPλ((z1, … , zn), (w1, … , wm); θ) with respect to a natural positive semi-definite, but degenerate, Hermitian product ‹·, ·›'n,m,θ. In case m = 0 (or n = 0), our product reduces to Macdonald's well-known inner product ‹·, ·›'n,θ, and we recover his corresponding orthogonality results for the Jack polynomials Pλ((z1, …, zn); θ). From our main results, we readily infer that the kernel of ‹·, ·›'n,m,θ is spanned by the super-Jack polynomials indexed by a partition λ not containing the m × n rectangle (mn). As an application, we provide a Hilbert space interpretation of the deformed trigonometric Calogero–Moser–Sutherland operators of type A(n − 1,m − 1).