The purpose of this paper is to give explicit symplectic iso morphisms from the cotangent bundle of the complex generalized flag var ieties onto the complex coadjoint orbits. If we let the complex Lie group G act on the cotangent bundle by an affine transformation instead of the canonical one, we shall see that the isomorphism is G-equivariant.

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A twisted moment map and its equivariance

1 Background on Differential Geometry.- 1.1 Fibre Bundles and Connections.- 1.2 Riemannian Manifolds.- 1.3 Superspaces.- 1.4 Superconnections.- 1.5 Characteristic Classes.- 1.6 The Euler and Thorn Classes.- 2 Asymptotic Expansion of the Heat Kernel.- 2.1 Differential Operators.- 2.2 The Heat Kernel on Euclidean Space.- 2.3 Heat Kernels.- 2.4 Construction of the Heat Kernel.- 2.5 The Formal Solution.- 2.6 The Trace of the Heat Kernel.- 2.7 Heat Kernels Depending on a Parameter.- 3 Clifford Modules and Dirac Operators.- 3.1 The Clifford Algebra.- 3.2 Spinors.- 3.3 Dirac Operators.- 3.4 Index of Dirac Operators.- 3.5 The Lichnerowicz Formula.- 3.6 Some Examples of Clifford Modules.- 4 Index Density of Dirac Operators.- 4.1 The Local Index Theorem.- 4.2 Mehler's Formula.- 4.3 Calculation of the Index Density.- 5 The Exponential Map and the Index Density.- 5.1 Jacobian of the Exponential Map on Principal Bundles.- 5.2 The Heat Kernel of a Principal Bundle.- 5.3 Calculus with Grassmann and Clifford Variables.- 5.4 The Index of Dirac Operators.- 6 The Equivariant Index Theorem.- 6.1 The Equivariant Index of Dirac Operators.- 6.2 The Atiyah-Bott Fixed Point Formula.- 6.3 Asymptotic Expansion of the Equivariant Heat Kernel.- 6.4 The Local Equivariant Index Theorem.- 6.5 Geodesic Distance on a Principal Bundle.- 6.6 The heat kernel of an equivariant vector bundle.- 6.7 Proof of Proposition 6.13.- 7 Equivariant Differential Forms.- 7.1 Equivariant Characteristic Classes.- 7.2 The Localization Formula.- 7.3 Bott's Formulas for Characteristic Numbers.- 7.4 Exact Stationary Phase Approximation.- 7.5 The Fourier Transform of Coadjoint Orbits.- 7.6 Equivariant Cohomology and Families.- 7.7 The Bott Class.- 8 The Kirillov Formula for the Equivariant Index.- 8.1 The Kirillov Formula.- 8.2 The Weyl and Kirillov Character Formulas.- 8.3 The Heat Kernel Proof of the Kirillov Formula.- 9 The Index Bundle.- 9.1 The Index Bundle in Finite Dimensions.- 9.2 The Index Bundle of a Family of Dirac Operators.- 9.3 The Chern Character of the Index Bundle.- 9.4 The Equivariant Index and the Index Bundle.- 9.5 The Case of Varying Dimension.- 9.6 The Zeta-Function of a Laplacian.- 9.7 The Determinant Line Bundle.- 10 The Family Index Theorem.- 10.1 Riemannian Fibre Bundles.- 10.2 Clifford Modules on Fibre Bundles.- 10.3 The Bismut Superconnection.- 10.4 The Family Index Density.- 10.5 The Transgression Formula.- 10.6 The Curvature of the Determinant Line Bundle.- 10.7 The Kirillov Formula and Bismut's Index Theorem.- References.- List of Notation.

Heat Kernels and Dirac Operators

Introduction I. FOUNDATIONS 1. From classical to modern 2. Linear symplectic geometry 3. Symplectic manifolds 4. Almost complex structures II. SYMPLECTIC MANIFOLDS 5. Symplectic group actions 6. Symplectic fibrations 7. Constructing symplectic manifolds III. SYMPLECTOMORPHISMS 8. Area-preserving diffeomorphisms 9. Generating functions 10. The group of symplectomorphisms IV. SYMPLECTIC INVARIANTS 11. The Arnold conjecture 12. Symplectic capacities 13. New directions

https://tspace.library.utoronto.ca/bitstream/1807/9754/1/symplectic_topology_1995.pdf

background

Introduction to symplectic topology

The Topology of Fibre Bundles. (PMS-14), Volume 14

In this classic work, Anthony W. Knapp offers a survey of representation theory of semisimple Lie groups in a way that reflects the spirit of the subject and corresponds to the natural learning process. This book is a model of exposition and an invaluable resource for both graduate students and researchers. Although theorems are always stated precisely, many illustrative examples or classes of examples are given. To support this unique approach, the author includes for the reader a useful 300-item bibliography and an extensive section of notes.

Representation Theory of Semisimple Groups: An Overview Based on Examples

0. The Capelli identity [Cal-3; W, p. 39] is one of the most celebrated and useful formulas of classical invariant theory [W; D; CL; Z]. The double commutant theorem [W, p. 91] is likewise a basic result in the general theory of associative algebras. Both play key roles in Weyl's book: The classical groups. The main purpose of this paper is to demonstrate a close connection between the two, in the context of multiplicity-free actions EKJ of groups on vector spaces. The focus of our discussion will be the structure of the differential operators which commute with a multiplicity-free action. Applications include a new derivation of some formulas of Shimura [Shl ] and Rubenthaler and Schiffmann [RS] for b-functions associated to Hermitian symmetric spaces, and a construction of interesting sets of generators for the center of the universal enveloping algebra of gin. We also give a detailed discussion of certain aspects of multiplicity-free representations. Here is an overview of the contents of the paper. In Sect. 1 we review the classical Capelli identities. We observe how their existence is predicted by a double commutant result (1.6) and (1.8), and we show how they can be used to compute b-functions. We remark that Capelli understood that his operators generate the center of the algebra generated by the polarization operators (see [B1, p. 77]). Also Capelli's motivation in introducing his operators was to "explain" a formula of Cayley essentially the computation of a b-function. Capelli's point of view was remarkably modern and structuralist, in certain ways more modern even than that of Weyl. Sections 2 through 9 provide a conceptual context for understanding the Capelli identities as a feature of multiplicity-free actions. They contain a general discussion of the structure of ~ G , the algebra of polynomial coefficient differential operators which commute with a given group G of linear transformations. The results of the discussion are summarized in Theorem 9.1, which says in particular that the polynomial coefficient differential operators commuting with a

A twisted moment map and its equivariance

References

Heat Kernels and Dirac Operators

Introduction to symplectic topology

The Topology of Fibre Bundles. (PMS-14), Volume 14

Representation Theory of Semisimple Groups: An Overview Based on Examples

The Capelli identity, the double commutant theorem, and multiplicity-free actions

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