**Contemporary Mathematics**

Volume: 610;
2014;
370 pp;
Softcover

MSC: Primary 17; 22;

Print ISBN: 978-0-8218-9170-4

Product Code: CONM/610

List Price: $126.00

AMS Member Price: $100.80

MAA Member Price: $113.40

**Electronic ISBN: 978-1-4704-1523-5
Product Code: CONM/610.E**

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AMS Member Price: $100.80

MAA Member Price: $113.40

# Perspectives in Representation Theory

Share this page *Edited by *
*Pavel Etingof; Mikhail Khovanov; Alistair Savage*

This volume contains the proceedings of the conference Perspectives in
Representation Theory, held from May 12–17, 2012, at Yale University,
in honor of Igor Frenkel's 60th birthday.

The aim of the conference was to present current progress on the
following (interrelated) topics: vertex operator algebras and chiral
algebras, conformal field theory, the (geometric) Langlands program,
affine Lie algebras, Kac-Moody algebras, quantum groups, crystal bases
and canonical bases, quantum cohomology and K-theory, geometric
representation theory, categorification, higher-dimensional Kac-Moody
theory, integrable systems, quiver varieties, representations of real
and \(p\)-adic groups, and quantum gauge theories.

The papers in this volume present representation theory connections
to numerous other subjects, as well as some of the most recent
advances in representation theory, including those which occurred
thanks to the application of techniques in other areas of mathematics,
and of ideas of quantum field theory and string theory.

#### Readership

Graduate students and research mathematicians interested in representation theory.

# Table of Contents

## Perspectives in Representation Theory

- Preface ix10 free
- On the work of Igor Frenkel 112 free
- Introduction by Pavel Etingof 112
- 1. Representation theory of affine Lie algebras by Alistair Savage and Anthony Licata 112
- 2. Quantum Knizhnik-Zamolodchikov equations by Pavel Etingof 314
- 3. Double loop groups by Pavel Etingof 415
- 4. Vertex operator algebras by John Duncan 617
- 5. Three-dimensional quantum gravity by John Duncan 718
- 6. Quaternionic analysis by Matvei Libine 920
- 7. Emergence of a new area –elliptic hypergeometric series by Michael Schlosser 1122
- 8. Representation theory of split real quantum groups and modular doubles by Ivan Ip 1223
- 9. Categorification by Mikhail Khovanov 1324
- 10. Geometric representation theory by Anthony Licata 1425
- References 1627

- Macdonald polynomials, Laumon spaces and perverse coherent sheaves 2334
- 1. Introduction 2334
- 2. Combinatorial notations 2738
- 3. Macdonald function 2940
- 4. Vanishing 3142
- 5. Parabolic Laumon spaces 3243
- 6. Euler characteristics of twisted De Rham complexes 3344
- 7. Difference equations 3748
- 8. Speculations for arbitrary simple groups 3950
- Acknowledgments 4051
- References 4152

- An affine Gindikin-Karpelevich formula 4354
- 1. Introduction 4354
- 2. Affine Kac-Moody groups 4859
- 3. Finiteness of the Gindikin-Karpelevich Sum, Part I 5061
- 4. Completion of 𝑈⁻ 5263
- 5. Finiteness of the Gindikin-Karpelevich Sum, Part 2 5566
- 6. Proof of Theorem 1.9 and Theorem 1.13 6071
- 7. Appendix: Proof of 5.9 6273
- Acknowledgements 6374
- References 6374

- On the discrete groups of Mathieu Moonshine 6576
- The classical master equation 7990
- 1. Introduction 7990
- 2. Graded varieties 8495
- 3. Solutions of the classical master equation 91102
- 4. Existence and uniqueness for affine varieties 93104
- 5. Computing the BRST cohomology 101112
- 6. Examples 104115
- 7. BRST cohomology in degree 0 and 1 and Lie–Rinehart cohomology 110121
- 8. The case of quasi-projective varieties 115126
- Appendix A. Stable isomorphism of Tate resolutions 119130
- Appendix B. Gluing sheaves of differential graded algebra 121132
- Acknowledgements 135146
- References 136147

- DG indschemes 139150
- Introduction 139150
- Acknowledgments 148159
- 1. DG indschemes 148159
- 2. Sheaves on DG indschemes 158169
- 3. Closed embeddings into a DG indscheme and push-outs 172183
- 4. Deformation theory: recollections 184195
- 5. A characterization of DG indschemes via deformation theory 195206
- 6. Formal completions 206217
- 7. Quasi-coherent and ind-coherent sheaves on formal completions 217228
- 8. Formally smooth DG indschemes 229240
- 9. Classical vs. derived formal smoothness 236247
- 10. 𝑄𝐶𝑜ℎ and 𝐼𝑛𝑑𝐶𝑜ℎ on formally smooth indschemes 248259
- References 251262

- Special Functions in Minimal Representations 253264
- Asymptotic Hecke algebras and involutions 267278
- Chiral differential operators on abelian varieties 279290
- Part 1. CHIRAL FOURIER-MUKAI TRANSFORM 281292
- Part 2. BOSON COMPACTFIED ON A TORUS 287298
- Refined Chern-Simons theory and Hilbert schemes of points on the plane 305316
- Categorified Jones-Wenzl Projectors: a comparison 333344
- Weil representations and theta functionals on surfaces 353364