The Capelli Identity for Grassmann Manifolds
نویسنده
چکیده
The column space of a real n× 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 G of k-planes in Rn, and hence a GLn-equivariant isomorphism C∞ (G) ≈ C∞ (X)k . We consider the On ×GLk-invariant differential operator C on X given by C = det ( xx ) det ( ∂∂ ) , where x = (xij) , ∂ = ( ∂ ∂xij ) . By the above isomorphism C defines an On-invariant operator on G. Since G is a symmetric space for On, the irreducible On-submodules of C∞ (G) have multiplicity 1; thus On-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 [9] and also gives new Capelli-type identities for the orthogonal Lie algebra.
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