Structure and properties of Lie group representations

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Note: Numbered references refer to my vita.

A representation of a group is a homomorphism into a group of unitary operators on some Hilbert space. Additionally, one typically imposes some extra technical conditions, but I will ignore these for the present purposes. The building blocks for studying the general representation theory come in two different flavors: irreducible representations and induced representations. Irreducible representations are fundamental in the sense that the group leaves no subspace of the underlying Hilbert space invariant. Irreducible representations are easily defined, but more difficult to study, construct or classify. In contrast, induced representations are easy to construct, modeled on the idea that a group naturally acts on the sections of a homogeneous vector bundle. Induced representations may not be irreducible, but they are pieced together from a finite number of irreducible representations.

My work with Luis Casian in [16] offers a fundamental new approach to studying the structure and properties of induced representations. The techniques used are of a geometric nature and involve describing the so-called weight filtration for an induced representation. This tells us how the various irreducible pieces fit together inside the induced representation. The results of this paper were made explicit in a special case in an earlier book [7] and the AMS Memoir [18], where the so-called real rank one case is discussed; this is an important testing ground for ideas and conjectures. My work with Brian Boe in the AMS Memoir [26] sheds new light on the structure of a particularly important class of induced representations and makes progress on a long standing problem of Vogan. (Problem #3 in the back list of problems in his book Representations of real reductive Lie groups, Birkhauser 1981. ) The work in [6] focuses on describing irreducible representations where an analog of Kostant's famous cohomology result (used to prove the Borel-Weil-Bott Theorem) holds.

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