Fall, 2008

• Speaker Paul Baum, Penn State
  • Title Geometric structure in the representation theory of reductive p-adic groups
  • Time/place Friday, 9/5/2008, 11:45 am in Hill 525
  • Abstract Let G be a reductive p-adic group. Examples are GL(n, F) SL(n, F) etc where n is any positive integer and F is any finite extension of the p-adic numbers Q_p. G^ denotes the admissible dual of G, i.e. the set of equivalence classes of smooth irreducible representations of G. Contained within G^ is the tempered dual of G, i.e. the set of equivalence classes of smooth irreducible representations of G having tempered Harish-Chandra character. G^ is in bijection with Prim(HG) where HG denotes the Hecke algebra of G. Thus HG is the convolution algebra of all locally-constant compactly- supported complex-valued functions f : G ---> C. Prim(HG) denotes the set of primitive ideals in HG. If Prim(HG) is given the Jacobson topology, Prim(HG) is then the disjoint union of its connected components. These connected components are known as the Bernstein components. This talk explains a conjecture due to A.M.Aubert, P.F.Baum and R.J.Plymen. According to the conjecture, each Bernstein component is a complex affine variety. These varieties are explicitly identified as extended quotients. The conjecture is based on the theory of the Bernstein center, and if correct further develops this theory. The talk is intended for non-specialists. All the basic definitions will be carefully stated.

• Speaker Minxian Zhu, Rutgers
  • Title Vertex algebras of differential operators over algebraic groups
  • Time/place Friday, 9/26/2008, 11:45 am in Hill 525
  • Abstract I will talk about a family of vertex algebras arising from an attempt to extend the classical algebra of differential operators on a Lie group to a vertex algebra. The talk is meant to be introductory. I will discuss various motivations and constructions. The goal is to convey the idea instead of technical details.

• Speaker Jim Lepowsky, Rutgers
  • Title Logarithmic tensor product theory---an -- --introduction
  • Time/place Friday, 10/10/2008, 11:45 am in Hill 525
  • Abstract I will give a non-technical introduction to joint work with Yi-Zhi Huang and Lin Zhang, in which we construct canonical braided tensor category structure on a suitable module category for a vertex operator algebra. The modules are not assumed completely reducible, and this entails the systematic use of logarithmic intertwining operators.

• Speaker Haisheng Li, Rutgers-Camden
  • Title Quantum Weyl-Clifford algebras and quantum vertex algebras
  • Time/place Friday, 10/17/2008, 11:45 am in Hill 525
  • Abstract In the quantum world, a family of associative algebras, called quantum Weyl-Clifford algebras, naturally arises from the study of noncommutative quantum field theory and noncommutative algebraic geometry. This family of algebras is also closely related to Zamolodchikov-Faddeev algebras. Just like Weyl algebras and Clifford algebras, quantum Weyl-Clifford algebras are of great interest from various points of view. In particular, we are interested in these quantum algebras from a vertex-algebra point of view; we are very much interested in canonical associations with vertex algebra-like structures. In this talk, we shall discuss certain (simpler) quantum Weyl-Clifford algebras associated with multiplicative skew matrices. Then we use them to construct quantum vertex algebras and modules. This talk will be introductory and knowledge of vertex algebras is not required.

• Speaker Tom Robinson, Rutgers
  • Title Equivalent axioms for a vertex algebra without vacuum
  • Time/place Friday, 10/24/2008, 11:45 am in Hill 525
  • Abstract Vertex-type algebras may be defined in a variety of ways. We shall use a formal calculus approach. However, even within the formal calculus setting there is a variety of major and minor axioms. Some of these replacement axioms have been found very useful in the representation theory of and the construction of vertex algebras. To simplify the issue we shall use a very minimal set of minor axioms. That is, we shall not include either the conformal or vacuum vectors. Omission of a vacuum vector, for instance, is analogous to working without an identity in ring theory. (No, we will not call the resulting algebra an "ertex algebra.") We shall first review in detail the standard definition of a vertex algebra. Then we shall derive some of the usual replacement axioms while in addition introducing a new one which simplifies some earlier proofs. The entire discussion will also be a self-contained introduction to typical computations used in the formal calculus.

• Speaker Steve Miller, Rutgers
  • Title Rapid decay of automorphic forms
  • Time/place Friday, 10/31/2008, 11:45 am in Hill 525
  • Abstract A fundamental result in automorphic forms, usually attributed to Gelfand and Piatetski-Shapiro, asserts the rapid decay of cusp forms on arithmetic quotients of semisimple Lie groups. This proves that certain integrals of them converge, and have analytic continuations; this is the analytic basis of the Rankin-Selberg method, which produces the analytic properties of some Langlands L-functions from such integrals. Schmid and I have developed an alternative method to Rankin-Selberg using distributions, which rests on the analyticity of invariant multilinear pairings of automorphic distributions. I will discuss this analyticity and its connection to the classical rapid decay results.

• Speaker Vladimir Retakh, Rutgers
  • Title Theory of Noncommutative Stochastic Matrices
  • Time/place Friday, 11/7/2008, 11:45 am in Hill 525
  • Abstract I will state basic results and discuss some applications.

• Speaker Ting Xue, MIT
  • Title Nilpotent orbits in characteristic 2 and the Springer correspondence
  • Time/place Friday, 11/14/2008, 11:45 am in Hill 525
  • Abstract Let G be an adjoint algebraic group of type B, C or D defined over an algebraically closed field k of characteristic 2 and g be the Lie algebra of G. Let g^* be the dual vector space of g. We classify the nilpotent orbits in g over the finite field F_{2^N} and construct a Springer correspondence for the nilpotent variety in g. The correspondence would be a bijective map between the set of isomorphism classes of irreducible representations of the Weyl group of G and the set of all pairs (c, F), where c is a nilpotent G-orbit in g and F is an irreducible G-equivariant local system on $\mathrm{c}$ (up to isomorphism). We also classify the nilpotent orbits in g^* over k and over F_{2^N} and construct a Springer correspondence for g^*.

• Speaker Leon Ehrenpreis, Temple and Rutgers
  • Title Partial differential equations and SL(3,R)
  • Time/place Friday, 11/21/2008, 11:45 am in Hill 525
  • Abstract Given a linear group and a finite dimensional representation there is an associated system of partial differential equations. This system of pde can be used to decompose the representation on the solutions of the system of pde to obtain irreducible representations. In the case of SL(3,R) we can use these methods to construct automorphic forms and Eisenstein and Poincare series.

• Speaker Rami Aizenbud and Dima Gurevitch, Weizmann Institute
  • Title Multiplicity one Theorems (p-adic and real)
  • Time/place Friday, 12/5/2008, 11:45 am and 2:00 pm in Hill 525
  • Abstract Let F be a local field of characteristic 0. We consider distributions on GL(n+1,F) which are invariant under the adjoint action of GL(n,F). We prove that such distributions are invariant under transposition. This implies that an irreducible representation of GL(n+1,F), when restricted to GL(n,F) "decomposes" with multiplicity one. Similar theorems hold for orthogonal and unitary groups. In the first hour we will tell the proof for non-archimedean F. In the second hour we will tell the proof for archimedean F. Those proofs have similar parts but there are important differences. The advantage in the non-archimedean case is that a distribution supported on a closed subset is the same as a distribution defined on that closed subset, unlike the archimedean case. The advantage in the archimedean case is that we can use differential operators and hence the theory of D-modules. The results for non-archimedean F are a joint work with S. Rallis and G. Schiffmann. The results for archimedean F were proven independently, simultaneously and in a different way by Sun Binyong and Chen Bo Zhu.