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).

81R12 (secondary)

33C52 (primary)


Farrokh Atai

Kobe University

Martin Hallnäs

Chalmers, Mathematical Sciences, Analysis and Probability Theory

Edwin Langmann

Royal Institute of Technology (KTH)

Bulletin of the London Mathematical Society

0024-6093 (ISSN) 1469-2120 (eISSN)


Subject Categories

Algebra and Logic


Mathematical Analysis


Basic sciences



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