Conley: Talks

Applications of representation theory to quantization, Workshop on Equivariant Quantization, University of Luxembourg, July 2011

Abstract: We will describe some applications of the representation theory of finite dimensional simple Lie algebras to quantization over Euclidean manifolds. We will discuss the projective quantization of differential operator modules of Vect(R^m), and in odd dimensions, the conformal quantization of the restriction of these modules to the subalgebra of contact vector fields.

We will show how the Casimir operators can be used to detect resonant cases, how Clebsch-Gordan rules can be used to classify the affine-invariant maps and hence to construct the quantization in the non-resonant cases, and how lowest weight vectors help to compute the quantized vector field actions.

Invariant differential operators, Red River Conference on Automorphic Forms and Representation Theory, University of Oklahoma, October 2010

Abstract: We will describe Helgason's method of computing algebras of invariant differential operators on homogeneous spaces, using various Jacobi group actions as examples. Casimir operators and factorizations into covariant operators will be included. We will do our best not to require any knowledge of Lie theory.

Quantizations of differential operator modules, AMS Winter Meeting, Integrability of Dynamical Systems Special Session, San Francisco, January 2010

Abstract: Suppose that one has an infinite dimensional Lie algebra of vector fields on a manifold, for example the set of all vector fields, or, if the manifold has a contact structure, the contact vector fields. Assume that this Lie algebra contains a distinguished finite dimensional semisimple maximal subalgebra, usually called its projective or conformal subalgebra.

There are various spaces of differential operators on the manifold which carry natural representations of such Lie algebras. The projective or conformal quantization of such a representation is its decomposition into irreducible representations of the subalgebra. We discuss recent results on quantizations and their applications to cohomology, geometric equivalences and symmetries of differential operator modules, and indecomposable modules. Our methods are algebraic: we consider only Euclidean manifolds.

Vect(R) & x to the x to the x to the ..., two VIGRE Student Colloquiua, Louisiana State University, November 2009

Extremal projectors, Representation Theory Seminar, Louisiana State University, November 2009

Abstract: Let g be a complex finite dimensional reductive Lie algebra. The extremal projector P(g) is an element of a certain formal extension of the enveloping algebra U(g) which projects representations in Category O to their highest weight vectors along their lower weight vectors, provided that the denominator of P(g) does not act by zero. (This denominator is a formal product in U(h), h being the chosen Cartan subalgebra.)

In 1971 Asherova-Smirnov-Tolstoi discovered a noncommutative finite factorization of P(g), and in 1993 Zhelobenko discovered a commutative infinite product formula. We will discuss these results and some more recent formulas for the relative projector P(g,l), the projection to the highest l-subrepresentations, l being a Levi subalgebra. (The infinite commutative factorization of P(g,l) is known, but its denominator, a formal product in the center of U(l), is only known in a few simply laced cases. There are only preliminary results concerning finite factorizations of P(g,l).)

Modules of differential operators for vector field Lie algebras, AMS Sectional Meeting, Representation Theory Special Session, Waco, Texas, October 2009

Abstract: The Lie algebra Vec(R^m) of vector fields on Euclidean space contains the projective Lie algebra sl(m+1) as a maximal subalgebra. The space of differential operators on R^m is naturally a module under Vec(R^m). In this talk we will discuss the decomposition of this module under the projective subalgebra, and the use of this decomposition in analyzing the action of Vec(R^m). We will also mention some generalizations: the module of differential operators can be generalized to modules of differential operators between arbitrary tensor field modules, and in odd dimensions, Vec(R^m) can be replaced by the Lie algebra of contact vector fields, in which case the projective subalgebra is replaced by the conformal subalgebra, a copy of sp(m+1).