Research Seminar in Mathematics (MTH G450 - 30757)
Organized by Professor Jonathan Weitsman
Guest Speaker: Gregg Zuckerman
Yale University
Title: Harmonic Algebra
Date: Tuesday, January 27, 2009
Time: 1:00 p.m.
Location: 509 Lake Hall
Pretalk I (1:00 - 2:00):Abstract: Harmonic algebra is my favorite term for a synthesis of my two mathematical interests, harmonic analysis and homological algebra. Harmonic analysis is a branch of functional analysis and incorporates the theory of unitary representations of groups on possibly infinite dimensional Hilbert spaces. Homological algebra is a branch of abstract algebra and algebraic topology and incorporates the theory of complexes of possibly infinite dimensional modules over possibly infinite dimensional associative algebras. Even if the base field is the complex numbers, it is hardly obvious how to relate homological algebra to unitary representation theory. Harish-Chandra was the first to link the theory of unitary representations of a connected semisimple Lie group G to the theory of infinite dimensional modules over the universal enveloping algebra U g of the Lie algebra g of G. We will describe this link in great detail. Harish-Chandra’s theory leads to the notion of a unitarizable U g-module. We will then generalize to the notion of a Hermitian U g-module, for which the inner product is possibly indefinite. There is still no classification of
the irreducible unitarizable U g-modules for general G, but there is a classification for the irreducible Hermitian U g-modules. Thus, the description of the unitary irreducible representations of G is now reduced to the following problem: give a criterion for an irreducible Hermitian U g-module to have a positive definite inner product. Zuckerman, Vogan, Wallach and Knapp-Vogan have given sufficient conditions for an irreducible Hermitian U g-module to be positive definite. These conditions are formulated in terms of the method of cohomological induction, which we will describe in more detail in the research talk. In brief, cohomological induction involves the systematic construction of U g-equivariant Hermitian complexes, which are formal analogs of the de Rham complex of a smooth orientable compact manifold.
Break (2:00 - 2:30)
Research Talk (2:30 - 3:00):Research Talk (2:30 - 3:30)- Harmonic algebra is my favorite term for a synthesis of my two main mathematical interests, harmonic analysis and homological algebra. The most visible concept in harmonic algebra is cohomological induction, which is a key algebraic tool for the construction of infinite dimensional irreducible unitary representations of a real semisimple Lie group G. However, unlike Mackey’s concept of unitary induction in ordinary harmonic analysis, cohomological induction is not a functor from unitary representations of an arbitrary closed subgroup of G to unitary representations of G. Instead, a cohomological induction functor is defined on an appropriate category of modules over the Lie algebra l of a suitable Lie subgroup L of G and takes values in a category of graded modules over the Lie algebra g of G. The subgroup L is typically the centralizer in G of a compact torus in G, so that G/L can be realized as a special coadjoint orbit in the dual of g. Even if the module over the Lie algebra l arises from a unitary representation of L, the corresponding cohomological induction module over the Lie algebra g of G will not in general arise from a unitary representation of G. However, Vogan, Wallach and Knapp-Vogan have given very general sufficient conditions for a cohomological induction functor to produce a unitarizable module over g from a unitarizable module over the Lie algebra of l. The Atlas of Lie Groups Pro ject hopes to employ the current theory of cohomological induction in an ambitious program to reduce the study of the full unitary dual of G to a finite calculation. The announcement in March, 2007, of a successful super-computer computation for the split real form of E8 is a key step by the Atlas Pro ject towards the determination of the as yet unknown unitary dual of this maximally complicated real semisimple Lie group. The Atlas Pro ject will also deal with all other semisimple Lie groups (as long as the rank of the complexification of G is not too large.
Department tea (3:30 - 4:00)
Research Talk (4:00 - 5:00): Continued research talk.
Discussion at 5:00 p.m. followed by dinner with the speaker.
Previous Talks:
Fall 2008:9/16, 9/23, 10/07,10/28,11/18,12/02
Started: 23 September 2008 Last modified: 9 February 2009
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