# Acta Mathematica

Online ISSN: 1871-2509
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A new q-binomial theorem for Macdonald polynomials is employed to prove an A_n analogue of the celebrated Selberg integral. This confirms the g = A_n case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for every simple Lie algebra. \mathfrakg \mathfrak{g} .

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Let G be a locally compact abelian (henceforth abbreviated to "lea") group and let Aut(G) denote the full group of automorphisms of G with the g-topology. One asks "For which lea groups, G, is Aut(G) locally compact?" To answer this question in general would require a much more detailed understanding of the structure theory of lea groups than is presently available. However, in this paper we make an initial attack on the problem by answering the question for the case in which G contains a lattice non-trivially. We also give some partial results in the case that G contains a lattice trivially. We apply these results to the problem of determining those lea groups, G, for which the group, B(G), discussed by Weil in [16], is locally compact. More explicitly the contents of this paper are as follows. In w 2 we review the duality and structure theory of lea groups and establish notations. In w 3 we review the definition and properties of the g-topology including a general Asco]i theorem. In w 4 the main theorem is stated. The sufficiency and necessity of the main theorem are proved in items 5 and 6 respectively. In w 7 we give partial results on the question of the local compactness of Aut(G) in the case that G contains a lattice trivially. In w 8 the above results are applied to the question of the local compactness of B(G). Finally, w 9 is a counterexample, showing that the lattice hypothesis cannot be dropped. The results of this paper were included in the author's doctoral dissertation submitted to the Johns Hopkins University in June 1969. The author is pleased to have this opportunity to thank his thesis advisor, Professor J. I. Igusa, for posing the question treated herein, for his many helpful suggestions, and his many excellent lectures.

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by (G 0, G) system of imprimitivity we mean a system of imprimitivity for G o based on G, acting in some separable Hilbert space ~H. In this paper we show that each (N, K) system of imprimitivity (V, E) gives rise to an (R, B) system of imprimitivity (lY, ~). If U denotes the unitary group (indexed by/~ =F/F0) associated with E, and F denotes the spectral measure of V (defined on Borel subsets of T, the circle group), then (U, F) is a (/~, T) system of imprimitivity. We show that (U, F) gives rise in a natural way to a (F, R) system of imprimitivity (~, P), and that every (F, R) system of imprimitivity is equivalent to a system of imprimitivity (~, P). Finally if ~ denotes the unitary group indexed by F with spectral measure E and _F the spectral measure of 17, then (U, F) and (U, ~) are equivalent systems of imprimitivity. We thus complete the circle of ideas involved in Gamelin's work.

Article
We give upper and lower bounds for the order of the top Chern class of the Hodge bundle on the moduli space of principally polarized abelian varieties.

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Let f:C -> A be an entire holomorphic curve into a semi-Abelian variety A. Then the Zariski closure of f(C) is a translate of a semi-Abelian subvariety of A (logarithmic Bloch-Ochiai's theorem). The purpose of the present paper is to establish a quantitative version of the above result for such f i.e., the second main theorem and the defect relation. Comment: LaTeX, 37 pages

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• University of Lorraine