The Capelli identity for Grassmann manifolds

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.