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تسجيل مستخدم جديدQuadratic Capelli operators and Okounkov polynomials
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Let $Z$ be the symmetric cone of $r times r$ positive definite Hermitian matrices over a real division algebra $mathbb F$. Then $Z$ admits a natural family of invariant differential operators -- the Capelli operators $C_lambda$ -- indexed by partitions $lambda$ 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 F^n $ with $n geq 2r$. Using this we construct a family of invariant differential operators $D_{lambda,s}$ on $Y$ that we refer to as quadratic Capelli operators. Our main result shows that the eigenvalues of the $D_{lambda,s}$ are given by specializations of Okounkov interpolation polynomials.
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