In this paper we determine the structure of the natural \widetilde{U}(n,n) module \Ol which is the Howe quotient corresponding to the determinant character \det^l of U(p,q). We first give a ...description of the tempered distributions on M_{p+q,n}(\C) which transform according to the character \det^{-l} under the linear action of U(p,q). We then show that after tensoring with a character, \Ol can be embedded into one of the degenerate series representations of U(n,n). This allows us to determine the module structure of \Ol. Moreover we show that certain irreducible constituents in the degenerate series can be identified with some of these representations \Ol or their irreducible quotients. We also compute the Gelfand-Kirillov dimensions of the irreducible constituents of the degenerate series.
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Let (G,G') be the reductive dual pair (O(p,q),Sp(2n,\mathbb{R})). We show that if \pi is a representation of Sp(2n,\mathbb{R}) (respectively O(p,q)) obtained from duality correspondence with some ...representation of O(p,q) (respectively Sp(2n,\mathbb{R})), then its Gelfand-Kirillov dimension is less than or equal to (p+q)(2n-\frac{p+q-1}{2}) (respectively 2n(p+q-\frac{2n+1}{2})).
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This paper considers the boundary rigidity problem for a compact convex Riemannian manifold (M,g) with boundary \partial M whose curvature satisfies a general upper bound condition. This includes all ...nonpositively curved manifolds and all sufficiently small convex domains on any given Riemannian manifold. It is shown that in the space of metrics g' on M there is a C^{3,\alpha }-neighborhood of g such that g is the unique metric with the given boundary distance-function (i.e. the function that assigns to any pair of boundary points their distance --- as measured in M). More precisely, given any metric g' in this neighborhood with the same boundary distance function there is diffeomorphism \varphi which is the identity on \partial M such that g'=\varphi ^{*}g. There is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance-function.
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An elementary proof of the following result, due to Schaefer, Wolff, and Arendt is given: if $T$ is a lattice homomorphism on a Banach lattice $E$ with spectrum $\sigma(T) = \{1 \}$, then $T = I$, ...the identity mapping on $E$.
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