Quadratic Capelli operators  and Okounkov polynomials

Sahi, Siddhartha; Salmasian, Hadi

doi:10.24033/asens.2399

Quadratic Capelli operators and Okounkov polynomials
[Opérateurs de Capelli quadratiques et polynômes d'Okounkov]

Sahi, Siddhartha ; Salmasian, Hadi

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 4, pp. 867-890.

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Résumé
Abstract

Soit $Z$ le cône symétrique de matrices de tailles $r \times r$ hermitiennes positives sur une algèbre de division réelle $𝔽$ . Alors $Z$ admet une famille naturelle d'opérateurs différentiels invariants — les Opérateurs de Capelli $C_{λ}$ — indexés par des partitions $λ$ de longueur au plus $r$ , dont les valeurs propres sont des spécialisations de polynômes d'interpolation Knop-Sahi.

Dans cet article, nous considérons une double fibration $Y ⟵ X ⟶ Z$ où $Y$ est la variété grassmanienne des sous-espaces de dimension $r$ de $𝔽^{n}$ avec $n \geq 2 r$ . En utilisant cela, nous construisons une famille d'opérateurs différentiels invariants $D_{λ, s}$ sur $Y$ que nous appelons opérateurs de Capelli quadratiques. Notre résultat principal montre que les valeurs propres des $D_{λ, s}$ sont des spécialisations de polynômes d'interpolation Okounkov.

Let $Z$ be the symmetric cone of $r \times r$ positive definite Hermitian matrices over a real division algebra $𝔽$ . 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 specializations of Knop-Sahi interpolation polynomials.

In this paper we consider a double fibration $Y ⟵ X ⟶ Z$ where $Y$ is the Grassmanian of $r$ -dimensional subspaces of $𝔽^{n}$ with $n \geq 2 r$ . 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 specializations of Okounkov interpolation polynomials.

MR Zbl

DOI : 10.24033/asens.2399

Classification : 05E05, 22E46.
Keywords: Grassmannian manifolds, Harish-Chandra homomorphism, Okounkov polynomials, quadratic Capelli operators, symmetric cones.
Mot clés : Variétés grassmanniennes, homomorphisme de Harish-Chandra, polynômes d'Okounkov, opérateurs de Capelli quadratiques, cônes symétriques.

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     title = {Quadratic {Capelli} operators  and {Okounkov} polynomials},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {867--890},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 52},
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     year = {2019},
     doi = {10.24033/asens.2399},
     mrnumber = {4038454},
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Sahi, Siddhartha; Salmasian, Hadi. Quadratic Capelli operators  and Okounkov polynomials. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 4, pp. 867-890. doi : 10.24033/asens.2399. http://www.numdam.org/articles/10.24033/asens.2399/

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