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Representation Theory

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The Capelli identity for Grassmann manifolds

Author: Siddhartha Sahi
Journal: Represent. Theory 17 (2013), 326-336
MSC (2010): Primary 22E46, 43A90
Published electronically: June 7, 2013
MathSciNet review: 3063840
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Abstract: The column space of a real $ n\times k$ matrix $ x$ of rank $ k$ is a $ k$-plane. Thus we get a map from the space $ X$ of such matrices to the Grassmannian $ \mathbb{G}$ of $ k$-planes in $ \mathbb{R}^{n}$, and hence a $ GL_{n} $-equivariant isomorphism

$\displaystyle C^{\infty }\left ( \mathbb{G}\right ) \approx C^{\infty }\left ( X\right ) ^{GL_{k}}$$\displaystyle \text {.} $

We consider the $ O_{n}\times GL_{k}$-invariant differential operator $ C$ on $ X$ given by

$\displaystyle C=\det \left ( x^{t}x\right ) \det \left ( \partial ^{t}\partial \right ),$$\displaystyle \quad \text {where }x=\left ( x_{ij}\right ),\text { }\partial =\left ( \frac {\partial }{\partial x_{ij}}\right ). $

By the above isomorphism, $ C$ defines an $ O_{n}$-invariant operator on $ \mathbb{G}$.

Since $ \mathbb{G}$ is a symmetric space for $ O_{n}$, the irreducible $ O_{n}$-submodules of $ C^{\infty }\left ( \mathbb{G}\right ) $ have multiplicity 1; thus, $ O_{n}$-invariant operators act by scalars on these submodules. Our main result determines these scalars for a general class of such operators including $ C$. This answers a question raised by Howe and Lee and also gives new Capelli-type identities for the orthogonal Lie algebra.

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Siddhartha Sahi
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey

Keywords: Capelli identity, Grassmannian, invariant differential operator
Received by editor(s): April 28, 2012
Received by editor(s) in revised form: December 13, 2012
Published electronically: June 7, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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