# Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

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## The Capelli identity for Grassmann manifoldsHTML articles powered by AMS MathViewer

by Siddhartha Sahi
Represent. Theory 17 (2013), 326-336
DOI: https://doi.org/10.1090/S1088-4165-2013-00434-X
Published electronically: June 7, 2013

## 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 $C^{\infty }\left ( \mathbb {G}\right ) \approx C^{\infty }\left ( X\right ) ^{GL_{k}}\text {.}$ We consider the $O_{n}\times GL_{k}$-invariant differential operator $C$ on $X$ given by $C=\det \left ( x^{t}x\right ) \det \left ( \partial ^{t}\partial \right ),\quad \text {where }x=\left ( x_{ij}\right ),\text { }\partial =\left ( \frac {\partial }{\partial x_{ij}}\right ).$ 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.

References
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Bibliographic Information
• Siddhartha Sahi
• Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey
• MR Author ID: 153000
• Email: sahi@math.rutgers.edu
• Received by editor(s): April 28, 2012
• Received by editor(s) in revised form: December 13, 2012
• Published electronically: June 7, 2013