By Alexander Astashkevich (auth.), Jean-Luc Brylinski, Ranee Brylinski, Victor Nistor, Boris Tsygan, Ping Xu (eds.)

This publication is an outgrowth of the actions of the heart for Geometry and Mathematical Physics (CGMP) at Penn nation from 1996 to 1998. the heart was once created within the arithmetic division at Penn kingdom within the fall of 1996 for the aim of selling and aiding the actions of researchers and scholars in and round geometry and physics on the college. The CGMP brings many viewers to Penn nation and has ties with different study teams; it organizes weekly seminars in addition to annual workshops The publication includes 17 contributed articles on present study subject matters in a number of fields: symplectic geometry, quantization, quantum teams, algebraic geometry, algebraic teams and invariant concept, and personality­ istic periods. lots of the 20 authors have talked at Penn kingdom approximately their learn. Their articles current new effects or speak about fascinating perspec­ tives on fresh paintings. all of the articles were refereed within the general model of fine medical journals. Symplectic geometry, quantization and quantum teams is one major subject matter of the e-book. a number of authors examine deformation quantization. As­ tashkevich generalizes Karabegov's deformation quantization of Kahler manifolds to symplectic manifolds admitting transverse polarizations, and reviews the instant map with regards to semisimple coadjoint orbits. Bieliavsky constructs an specific star-product on holonomy reducible sym­ metric coadjoint orbits of an easy Lie workforce, and he exhibits tips to con­ struct a star-representation which has fascinating holomorphic properties.

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We choose X'I/J = Eo,n, x~ = Eo,o - En,n, and Xo = En,o. Then the normalized Killing form on 9 is given by (X,X')g = ~(TraceXX'). We take the Cartan involution a to be a(X) = -X*. The matrices x~ = En,p and xp = Ep,o, p = 1, ... , m, form a basis of 9-1 and satisfy the conditions in (33). So we get o 0 o o WI m w = '2:(wpEp,o + w~En,p) = p=1 0 o o (71) o o 0 w~ w~ 0 Next we need to compute the polynomial P defined in (48). We have the standard operator identity Ad(exptw) = iT(adtw)k and so (48) gives 2::0 P(Wi' w:) is the coefficient of t4xo in (Ad(exptw)) .

Unless X is smooth and affine, there is no guarantee that a given symbol will quantize. In this paper we quantize the symbols r x into order 4 differential operators Dx on 0 in a manner equivariant with respect to both the G-action and the Euler C* -action. We show that this equivariant quantization is unique. In our next paper [A-B3], we use these same operators Dx to quantize 0 by quantizing the map (6). We obtain a star product on R(O) given by "pseudo-differential" operators. We construct the operators Dx by manufacturing a single operator Do = Dxo E V~l(O) where Xo Egis a lowest weight vector.

In fact the quotient XjC* is the unique closed G-orbit in the projective space JP>(V). The quotient Xj(X) of X. The closure CI(X) of X in V is equal to Xu {O} and X is the unique non-zero minimal conical G-orbit in V. The orbit X has several very nice properties ([V-P]): (i) The closure CI(X) is normal. (ii) The graded G-equivariant algebra homomorphism R(CI(X)) given by restriction of functions is an isomorphism. ---+ R(X) (iii) The algebra R(X) is multiplicity-free as a G-representation and Rp(X) is G-isomorphic to the pth Cartan power of V*.

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