# Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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

by Siddhartha Sahi
Represent. Theory 17 (2013), 326-336 Request permission

## 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.

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