New Developments in Lie Theory and Their Applications

1. Front Matter

   Pages i-ix

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2. Automorphic Forms

   • Nolan R. Wallach

   Pages 1-25

3. Analytic and Geometric Realization of Representations

   • Wilfried Schmid

   Pages 27-47

4. Introduction to Quantized Enveloping Algebras

   • George Lusztig

   Pages 49-65

5. The Vanishing of Scalar Curvature on 6 Manifolds, Einstein’s Equation, and Representation Theory

   • Bertram Kostant

   Pages 67-86

6. Unitary Representations of Reductive Lie Groups and the Orbit Method

   • David A. Vogan Jr.

   Pages 87-114

7. Twistor Theory for Riemannian Manifolds

   • John Rawnsley

   Pages 115-128

8. You Can’t Hear the Shape of a Manifold

   • Carolyn S. Gordon

   Pages 129-146

9. Kuznetsov Formulas

   • R. J. Miatello

   Pages 147-154

10. Lefschetz Numbers and Cyclic Base Change for Purely Imaginary Extensions

    • Jurgen Rohlfs, Birgit Speh

    Pages 155-161

11. Some Zeta Functions Attached to Γ\G/K

    • Floyd L. Williams

    Pages 163-177

12. On the Centralizer of K in the Universal Enveloping Algebra of SO(n, 1) and SU(n, 1)

    • Juan A. Tirao

    Pages 179-186

13. On Spherical Modules

    • Nicolás Andruskiewitsch

    Pages 187-198

14. Generalized Weil Representations for SL(n,k), n odd, k a Finite Field

    • José Pantoja, Jorge Soto-Andrade

    Pages 199-206

15. Local Multiplicity of the Intersection of Lagrangian Cycles and the Index of Holonomic Modules

    • Alberto S. Dubson

    Pages 207-225

16. Back Matter

    Pages 227-228

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Representation theory, and more generally Lie theory, has played a very important role in many of the recent developments of mathematics and in the interaction of mathematics with physics. In August-September 1989, a workshop (Third Workshop on Representation Theory of Lie Groups and its Applications) was held in the environs of C6rdoba, Argentina to present expositions of important recent developments in the field that would be accessible to graduate students and researchers in related fields. This volume contains articles that are edited versions of the lectures (and short courses) given at the workshop. Within representation theory, one of the main open problems is to determine the unitary dual of a real reductive group. Although this prob­ lem is as yet unsolved, the recent work of Barbasch, Vogan, Arthur as well as others has shed new light on the structure of the problem. The article of D. Vogan presents an exposition of some aspects of this prob­ lem, emphasizing an extension of the orbit method of Kostant, Kirillov. Several examples are given that explain why the orbit method should be extended and how this extension should be implemented.

• algebra
• lie group
• representation theory

• Facultad de Matematica Astronomia y Fisica, Universidad Nacional de Córdoba, Córdoba, Argentina

  Juan Tirao

• Dept. of Mathematics, University of California at San Diego, La Jolla, USA

  Nolan R. Wallach

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