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 guts used to be 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 country and has ties with different study teams; it organizes weekly seminars in addition to annual workshops The ebook includes 17 contributed articles on present learn themes in quite a few fields: symplectic geometry, quantization, quantum teams, algebraic geometry, algebraic teams and invariant idea, and personality istic periods. many of the 20 authors have talked at Penn nation approximately their learn. Their articles current new effects or talk about attention-grabbing perspec tives on fresh paintings. the entire articles were refereed within the typical model of good clinical journals. Symplectic geometry, quantization and quantum teams is one major topic of the e-book. numerous authors research deformation quantization. As tashkevich generalizes Karabegov's deformation quantization of Kahler manifolds to symplectic manifolds admitting transverse polarizations, and reviews the instant map on the subject of semisimple coadjoint orbits. Bieliavsky constructs an specific star-product on holonomy reducible sym metric coadjoint orbits of an easy Lie team, and he indicates easy methods to con struct a star-representation which has fascinating holomorphic properties.

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E, ). = symbol E. (39) Then). is G-invariant. The canonical lift of the Euler C*-action on 0 defines a G-invariant algebra grading R(T*O) = $PEzRp(T*O). Then Rp(T*O) = { E R(T*O) I {,X,} = p}. (40) We say that is Euler homogeneous of degree p if E Rp{T*O). Now we get a G-invariant complex algebra bigrading R{T*O) = E9 pEZ,dE'4 ~(T*O), (41) 34 ALEXANDER ASTASHKEVICH AND RANEE BRYLINSKI where R'/,(T*O) = Rd(T*O) n Rp(T*O). The order d symbol mapping sends V:(O) into R:(T*O). , q,X = symbol rJx (42) for all x E g.

3, is to quantize the symbol S = foro into an operator S which satisfies condition (ii) in Prop. 4. To construct S and compute S(fj), we will simply work out everything in terms of our local coordinates fo, f~, Ii, fI on 0 from (43). 1 in [A-B2J, we first found a formula for the highest weight vector ft/J E R1(O) ~ g. 1. [A-B2] The unique expression for the function f,p E R1(O) in terms of our local coordinates fo, f~, fi, f:, i = 1, ... , m, on 0 is (63) The unique expressions for the vector fields "lXi, "Ix; ,TJxo in terms of our local coordinates fi, fI, fo, f~ are, where i = 1, ...

Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math, 43 (1971), 753-809 P. B. Kronheimer, Instantons and the geometry of the nilpotent variety, Jour. Diff. A. Lecomte and V. Ovsienko, Projectively invariant symbol map and cohomology of vector fields Lie algebras intervening in quantization, preprint [L-Sm] T. P. Smith, Primitive ideals and nilpotent orbits in type G 2 , J. Algebra, 114 (1988), 81-105 [L-Sm-St] T. P. T. Stafford, The minimal nilpotent orbit, the Joseph ideal and differential operators, J.