By Bloch S. (ed.)

ISBN-10: 082181480X

ISBN-13: 9780821814802

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Ucdavis . edu Ranee Brylinski Department of Mathematics Penn State University E-mail address: rkblDmath. psu. edu Received 11/23/97; revised 8/5/97 THE GEOMETRY SURROUNDING THE ARNOLD-LIOUVILLE THEOREM AUGUSTIN BANYAGA ABSTRACT. e. a fibration 7r : M2n -+ wn of a symplectic 2n-dimensional manifold M over an n manifold W with isotropic tori of various dimensions as fibers. This definition , which contains as particular cases, completely integrable hamiltonian systems, hamiltonian actions, and Duistermaat (lagrangian) fibrations, is extended to the contact category, and the following famous results: Arnold-Liouville theorem, Atiyah-Guillemin-Sternberg convexity of the moment map theorem, Delzant realization theorem, Duistermaat theory have been shown to admit a generalization to the contact category in the paper [6] to which this expository paper may serve as an introduction.

Then there is no non-zero G-linear map 9 ~ R~l (T*O). That is, R~l (T*O) contains no copy of the adjoint representation. 2. The formula (51) applies equally well when 9 = sp(2n, q, n 2: 1. But then P = o. The symbol fol( x o)2 easily quantizes to a . differential operator on o. See [A-B2]. Our main result is the G-equivariant quantization of these symbols r x into differential operators Dx on 0 in the cases where 9 is classical. 3. Assume 9 is a complex simple Lie algebra of classical type and 9 =1= sp(2n, q, n 2: 1.

The functions f;, k = 0,1,2 ... ,x. Precisely, f; generates Rk(O) under the action of n- so that U(n-) . f; = Rk(O). We have n- c go E9 gneg and also go . 1(iii). It follows that (61) Now suppose TU;) E R(O). ,Z for all Z E gneg, we get T(Rk(O)) = T(U(gneg) . f;) = U(gneg) . TU;) c R(O). 2. The converse is obvious. 4. Suppose S is a differential operator on oreg such that S is Euler homogeneous of degree and S satisfies (59). of~-l for k = 0,1,2 ... where ,0,/1, ... are scalars. Moreover (i) and (ii) imply (iii) fo1S is a lowest weight vector of a copy ofg in V_ 1(O) .

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Algebraic Geometry - Bowdoin 1985, Part 2 by Bloch S. (ed.)

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