C ONTEMPORARY M ATHEMATICS 557 Representation Theory and Mathematical Physics Conference in Honor of Gregg Zuckerman's 60th Birthday October 24–27, 2009 Yale University Jeffrey Adams Bong Lian Siddhartha Sahi Editors American Mathematical Society Representation Theory and Mathematical Physics C ONTEMPORARY M ATHEMATICS 557 Representation Theory and Mathematical Physics Conference in Honor of Gregg Zuckerman's 60th Birthday October 24–27, 2009 Yale University Jeffrey Adams Bong Lian Siddhartha Sahi Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss 2010 Mathematics Subject Classification. Primary22E45, 22E46,22E47, 17B65, 17B68, 17B69, 33D52. Library of Congress Cataloging-in-Publication Data Representationtheoryandmathematicalphysics: conferenceinhonorofGreggZuckerman’s60th birthday,October24–27,2009,YaleUniversity/JeffreyAdams,BongH.Lian,SiddharthaSahi, editors. p.cm. —(Contemporarymathematics;v.557) Includesbibliographicalreferences. ISBN978-0-8218-5246-0(alk.paper) 1. Linear algebraic groups—Congresses. 2. Representations of Lie groups—Congresses. 3. Mathematical physics—Congresses. I. Zuckerman, Gregg. II. Adams, Jeffrey. III. Lian, BongH.,1962- IV.Sahi,Siddhartha,1958- QA179.R47 2011 515(cid:2).7223—dc23 2011030808 Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansfor edu- cationaland scientific purposes without fee or permissionwith the exception ofreproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- ment of the source is given. 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(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 161514131211 Dedicated to Gregg Zuckerman on the occasion of his 60th birthday. This page intentionally left blank Contents Preface ix Expository Papers 1 The Plancherel Formula, the Plancherel Theorem, and the Fourier Transform of Orbital Integrals Rebecca A. Herb and Paul J. Sally, Jr. 3 Branching Problems of Zuckerman Derived Functor Modules Toshiyuki Kobayashi 23 Chiral Equivariant Cohomology of Spheres Bong H. Lian, Andrew R. Linshaw, and Bailin Song 41 Research Papers 77 Computing Global Characters Jeffrey Adams 79 Stable Combinations of Special Unipotent Representations Dan M. Barbasch and Peter E. Trapa 113 Levi Components of Parabolic Subalgebras of Finitary Lie Algebras Elizabeth Dan-Cohen and Ivan Penkov 129 On Extending the Langlands-Shahidi Method to Arithmetic Quotients of Loop Groups Howard Garland 151 The Measurement of Quantum Entanglement and Enumeration of Graph Coverings Michael W. Hero, Jeb F. Willenbring, and Lauren Kelly Williams 169 (cid:2) The Dual Pair (O ,OSp ) and Zuckerman Translation p,q 2,2 Dan Lu and Roger Howe 183 On the Algebraic Set of Singular Elements in a Complex Simple Lie Algebra Bertram Kostant and Nolan Wallach 215 An Explicit Embedding of Gravity and the Standard Model in E 8 A. Garrett Lisi 231 From Groups to Symmetric Spaces George Lusztig 245 vii viii CONTENTS Study of Antiorbital Complexes George Lusztig 259 Adelization of Automorphic Distributions and Mirabolic Eisenstein Series Stephen D. Miller and Wilfried Schmid 289 Categories of Integrable sl(∞)-, o(∞)-, sp(∞)- modules Ivan Penkov and Vera Serganova 335 Binomial Coefficients and Littlewood–Richardson Coefficients for Interpolation Polynomials and Macdonald Polynomials Siddhartha Sahi 359 Restriction of some Representations of U(p,q) to a Symmetric Subgroup Birgit Speh 371 Preface Lie groups and their representations are a fundamental area of mathematics, withconnectionstogeometry,topology,numbertheory,physics,combinatorics,and many other areas. Gregg Zuckerman’s work lies at the very heart of the modern theory of representations of Lie groups. His influential ideas on derived functors, thetranslationprinciple,andcoherentcontinuationlaidthegroundworkofmodern algebraic theory. Zuckerman has long been active in the fruitful interplay between mathemat- ics and physics. Developments in this area include work on chiral algebras, and the representation theory of affine Kac-Moody algebras. Recent progress on the geometric Langlands program points to exciting connections between automorphic representations and dual fibrations in geometric mirror symmetry. These topics were the subject of a conference in honor of Gregg Zuckerman’s 60th birthday, held at Yale, October 24-27, 2009. Summary of Contributions The classical Plancherel theorem is a statement about the Fourier transform on L2(R). It has generalizations to any locally compact group. The Plancherel Formula, The Plancherel Theorem, and the Fourier Transform of Orbital Integrals by Rebecca A. Herb and Paul J. Sally, Jr. surveys the history of this subject for non-abelian Lie groups and p-adic groups. One of Zuckerman’s major contributions to representation theory is the tech- nique now known as cohomological induction or the derived functor construction of representations. An important special case of this construction are the so-called A (λ) representations which are cohomologically induced from one-dimensional q characters. The paper Branching Problems of Zuckerman Derived Functor Mod- ules by Toshiyuki Kobayashi provides a comprehensive survey of known results on the restrictions of the A (λ) to symmetric subgroups, along with sketches of the q most important ideas of the proofs. Chiral Equivariant Cohomology of Spheres, by Bong H. Lian, Andrew R. Lin- shaw, and Bailin Song, is a survey of their work on the theory of chiral equivariant cohomology. This is a new topological invariant which is vertex algebra valued and contains the Borel-Cartan equivariant cohomology theory of a G-manifold as a substructure. The paper describes some of the general structural features of the new invariant—a quasi-conformal structure, equivariant homotopy invariance, and the values of this cohomology on homogeneous spaces—as well as a class of group actionsonsphereshavingthesameclassicalequivariantcohomology,butwhichcan all be distinguished by the new invariant. ix