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

Abstract

We prove orthogonality and compute explicitly the (quadratic) norms for super-Jack polynomials $S{P}_{\lambda }((z_1,\dots ,z_n),(w_1,\dots ,w_m);\theta )$ with respect to a natural positive semi-definite, but degenerate, Hermitian product $⟨\cdot ,\cdot {⟩}_{n,m}^{\prime }$. In case $m=0$ (or $n=0$), our product reduces to Macdonald's well-known inner product $⟨\cdot ,\cdot {⟩}_n^{\prime }$, and we recover his corresponding orthogonality results for the Jack polynomials $P_{\lambda }((z_1,\dots ,z_n);\theta )$. From our main results, we readily infer that the kernel of $⟨\cdot ,\cdot {⟩}_{n,m}^{\prime }$ is spanned by the super-Jack polynomials indexed by a partition $\lambda$ not containing the $m\times n$ rectangle $(m^n)$. As an application, we provide a Hilbert space interpretation of the deformed trigonometric Calogero-Moser-Sutherland operators of type $A(n-1,m-1)$.

19 pages, 1 figure