LIE THEORY WORKSHOP

Schedule for Saturday, March 5:

• 10:00 Coffee in room 104 (Lounge)
• 10:30 Karl-Hermann Neeb (Universität Darmstadt) : Triples in the Shilov boundary of bounded symmetric domains
  Abstract: If D is a finite-dimensional bounded symmetric domain of tube type and S its Shilov boundary, then the group G of automorphisms of D also acts on S and therefore on the set of triples in S. We report on a recent project with J.L. Clerc, where we classify the G-orbits in the space of triples in S. For the special cases where D is the open unit ball in the space of all n times n complex matrices, the Shilov boundary S is the group U(n), and this essentially amounts to the fact that triples of unitary matrices can be diagonlized simultaneously under the action of the group U(n,n), acting by fractional linear maps. The classification of triples leads in particular to an axiomatic description of the Maslov index introduced by J.L. Clerc and B. Oersted.

• 11:45 Lunch in room 104 (Lounge)
• 13:15 Alistair Savage (University of Toronto) : Quiver Varieties and Demazure Modules
  Abstract : Demazure modules are certain truncations of highest weight representations of Lie algebras. They yield a filtration of these (possibly infinite) representations by finite dimensional objects and have many uses in representation theory. In this talk we will discuss a new geometric realization of Demazure modules. We will use the quiver variety description of representations of Kac-Moody algebras developed by Lusztig and Nakajima and show that by restricting to certain subvarieties, the construction yields the Demazure modules. One benefit of this construction is that we see that Lusztig's semicanonical basis is compatible with the Demazure filtration. No prior knowledge of quiver varieties or Demazure modules will be assumed.

• 14:45 Wentang Kuo (University of Waterloo): Induced representations and nilpotent orbits
  Abstract: Let G be a connected split reductive p-adic group. In this talk, we explore the relation between principal nilpotent orbits of G and the irreducible constituents of the principal series of G. A geometric characterization of certain irreducible constituents is also provided. In addition, we can express the relation in terms of L-group objects.

• 15:45 Coffee in room 104 (Lounge)

• 16:15 Bruce Allison (University of Alberta) : Realizations of graded-simple algebras as loop algebras.
  Abstract : In the theory of infinite dimensional Lie algebras, the loop algebra construction gives realizations of all affine Kac-Moody Lie algebras. In this talk, we discuss a generalization of the loop algebra construction and show that it gives realizations of all graded-simple Z^n-graded algebras over an algebraically closed field of characteristic 0. We will describe applications of this result to the study of extended affine Lie algebras and quantum tori. The talk will be based on joint work with Stephen Berman, John Faulkner and Arturo Pianzola.

• 18:30 Conference dinner

Schedule for Sunday, March 6:

• 9:30 Coffee in room 104 (Lounge)
• 10:00 Michael Lau (University of Ottawa) : Bosonic and fermionic representations
  Abstract: Various Lie algebras have natural actions on highest weight modules for Weyl or Clifford algebras. These are called bosonic or fermionic representations. For affine Lie algebras, these include the vertex operator representations, as well as the oscillator/spinor modules. In this talk, I will describe a very general method for constructing such representations in a uniform manner. Special cases include some of the work done by A. Feingold and I.B. Frenkel (1985) for affine Lie algebras, and the results of Y. Gao (2002) for some extended affine Lie algebras.

• 11:15 David Vogan (MIT): The Kazhdan-Lusztig conjecture and signatures of invariant Hermitian forms
  Abstract: Suppose G is a real reductive group. The Kazhdan-Lusztig conjecture and the Jantzen conjecture (both of which have been proved) provide (almost) an algorithm for computing the signature of the invariant Hermitian form on any irreducible representation of G. Such an algorithm in particular decides whether the representation is unitary. The "almost" part is that certain signs in the algorithm are not known, so what the algorithm actually provides is a finite collection of possible formulas for the signature. On the other hand, signatures are dimensions, and are therefore non-negative; so one can discard possible formulas that lead to negative numbers.
  I will explain this algorithm a little more carefully, and examine some examples of the ambiguity in it.

Parking: Participants will be given a parking permit for the parking lot T behind the department. Please talk to one of the local organizers as soon as possible after your arrival.

Accommodations close to the department.

Getting around in Ottawa

Ottawa weather forecast

Financial Support: Financial support for participating graduate students and postdoctoral fellows is available. If you are interested please contact one of the local organizers as soon as possible, and ask your supervisor to do the same.

Registration: There is no registration and hence no registration fee for attending the workshop.

Local organizers: Erhard Neher (neher@uottawa.ca) and Wulf Rossmann(rossmann@uottawa.ca).