# Representation Theory and Mathematical Physics

Share this page *Edited by *
*Jeffrey Adams; Bong Lian; Siddhartha Sahi*

This volume contains the proceedings of the
conference on Representation Theory and Mathematical Physics, in honor
of Gregg Zuckerman's 60th birthday, held October 24–27, 2009, at Yale
University.

Lie groups and their representations play a fundamental role in
mathematics, in particular because of connections to geometry,
topology, number theory, physics, combinatorics, and many other
areas. Representation theory is one of the cornerstones of the
Langlands program in number theory, dating to the 1970s. Zuckerman's
work on derived functors, the translation principle, and coherent
continuation lie at the heart of the modern theory of representations
of Lie groups. One of the major unsolved problems in representation
theory is that of the unitary dual. The fact that there is, in
principle, a finite algorithm for computing the unitary dual relies
heavily on Zuckerman's work.

In recent years there has been a fruitful interplay between
mathematics and physics, in geometric representation theory, string
theory, and other areas. New developments on chiral algebras,
representation theory of affine Kac-Moody algebras, and the geometric
Langlands correspondence are some of the focal points of this
volume. Recent developments in the geometric Langlands program point
to exciting connections between certain automorphic representations
and dual fibrations in geometric mirror symmetry.

# Table of Contents

## Representation Theory and Mathematical Physics

- Contents vii8 free
- Preface ix10 free
- Expository Papers 114 free
- Research Papers 7790
- Computing Global Characters 7992
- Stable Combinations of Special Unipotent Representations 113126
- Levi Components of Parabolic Subalgebras of Finitary Lie Algebras 129142
- On Extending the Langlands-Shahidi Method to Arithmetic Quotients of Loop Groups 151164
- The Measurement of Quantum Entanglement and Enumeration of Graph Coverings 169182
- The Dual Pair (Op,q,OSp2,2) and Zuckerman Translation 183196
- On the Algebraic Set of Singular Elements in a Complex Simple Lie Algebra 215228
- An Explicit Embedding of Gravity and the Standard Model in E8 231244
- From Groups to Symmetric Spaces 245258
- 1. Notation. 246259
- 2. Almost diagonal symmetric spaces. 246259
- 3. A generalization of Schur's lemma. 246259
- 4. Elliptic curves arising from a symmetric space. 247260
- 5. Dimension of a nilpotent orbit. 247260
- 6. Some intersection cohomology sheaves. 248261
- 7. Generalization of a theorem of Steinberg. 248261
- 8. F-thin, F-thick nilpotent orbits. 248261
- 9. F-thin nilpotent orbits and affine canonical bases. 249262
- 10. Character sheaves on p. 249262
- 11. Computation of a Fourier transform. 250263
- 12. Another computation of a Fourier transform. 251264
- 13. Computation of a Deligne-Fourier transform. 252265
- 14. Nilpotent K-orbits and conjugacy classes in a Weyl group. 254267
- 15. Example: (GL2n, Sp2n). 255268
- 16. Example: (SO2n, SO2n−1). 256269
- 17. Example: (GL4,GL2 × GL2). 256269
- 18. Final comments. 258271

- Study of Antiorbital Complexes 259272
- Adelization of Automorphic Distributions and Mirabolic Eisenstein Series 289302
- Categories of Integrable sl(∞)-, o(∞)-, sp(∞)- modules 335348
- Binomial Coefficients and Littlewood–Richardson Coefficients for Interpolation Polynomials and Macdonald Polynomials 359372
- Restriction of some Representations of U(p, q) to a Symmetric Subgroup 371384