Preface Lie groups and their representations are a fundamental area of mathematics, with connections to geometry, topology, number theory, 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, the translation principle, and coherent continuation laid the groundwork of modern 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 Aq (λ) representations which are cohomologically induced from one-dimensional 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 Aq (λ) to symmetric subgroups, along with sketches of the 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 actions on spheres having the same classical equivariant cohomology, but which can all be distinguished by the new invariant. ix
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