Orthogonality of super‐Jack polynomials and a Hilbert space interpretation of deformed Calogero–Moser–Sutherland operators. Issue 2 (3rd February 2019)

• Record Type: Journal Article
• Authors: Atai, Farrokh; Hallnäs, Martin; Langmann, Edwin
• Is Part Of: Bulletin of the London Mathematical Society. Volume 51:Issue 2(2019)
• Page Start: 353
• Page End: 370
• Publication Date: 2019-02-03
• Subjects: 33C52 (primary) -- 81R12 (secondary); Mathematics -- Periodicals; 510
• DOI: 10.1112/blms.12234
• Languages: English
• ISSNs: 0024-6093

Abstract

Abstract: We prove orthogonality and compute explicitly the (quadratic) norms for super‐Jack polynomials S P λ ( ( z 1, …, z n ), ( w 1, …, w m ) ; 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 P λ ( ( z 1, …, z n ) ; 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 ( 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 ) .