Quadratic Capelli operators and Okounkov polynomials

2016/09/04 by Siddhartha Sahi, Sahi, Siddhartha, Hadi Salmasian +1
Computer Science · Mathematics · #05E05 #22E46 #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical functions and polynomials #Matrix Theory and Algorithms #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1609.00939

openalex publication_date 2016/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Z be the symmetric cone of r × r positive definite Hermitian matrices over a real division algebra \mathbb F. Then Z admits a natural family of invariant differential operators -- the Capelli operators C_λ -- indexed by partitions λ of length at most r, whose eigenvalues are given by specialization of Knop--Sahi interpolation polynomials. In this paper we consider a double fibration Y \longleftarrow X \longrightarrow Z where Y is the Grassmanian of r-dimensional subspaces of \mathbb Fⁿ with n ≥ 2r. Using this we construct a family of invariant differential operators D_λ,s on Y that we refer to as quadratic Capelli operators. Our main result shows that the eigenvalues of the D_λ,s are given by specializations of Okounkov interpolation polynomials.

Citations

• Positivity of Shimura operators

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