.)
My current research is in the area Representation Theory of Lie Groups. The study of representations is directly linked to harmonic analysis of functions on the underlying Lie group. For example, the classical treatment of Fourier analysis on the unit circle T depends on identifying the dual group, which parametrizes the irreducible unitary representations of T. More generally, for a group like Sl(2,R), an explicit Plancherel formula depends upon some understanding of irreducible unitary representations. Now the problem is much harder, since the group Sl(2,R) is highly non-commutative. However, the group Sl(2,R) is a Lie group and we can effectively exploit the interaction of the group structure and differential structure to analyze various function spaces.
Since every Lie group
is the semidirect product of a reductive group and a solvable group,
there is
a natural fork in the road when
studying the associated
representation theory and harmonic analysis.
My work has
focused on the setting of a reductive Lie group. Actually, everyone
is very aquainted with the notion of a reductive group, at least in
spirit, if not in name. To become convinced of this, check out my
review of Nolan Wallach's book
Real reductive groups I; this
review gives a readable
expository account of one aspect of the subject of
representation theory.
(But this link
is currently under construction .)
I work in four mainstream directions. I will highlight each below and you can link for more information if wish.
You can link to my vita to check out my various research publications. Most of these are research articles or memoirs, but there are two books: Representations of rank one Lie groups, 1985 and Nilpotent orbits in semisimple Lie algebras, 1993 (which I co-authored with Monty McGovern).
Click on picture for significance.
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 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 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.
Click on the formula to find out it's significance.
Highest weight representations arise naturally on the level of the associated Lie algebra. One can study these highest weight representations in isolation, in the context of modules over the enveloping algebra of a reductive Lie algebra. On the other hand, many problems in highest weight representation theory are either directly or indirectly linked to problems in Lie group representation theory. For example, Harish-Chandra built a family of unitary representations having discrete Plancherel measure which turn out to be highest weight representations; these are studied in [10]and [11]. For another example, the famous Kazhdan-Lusztig conjectures (1980) delt with describing the composition factors of highest weight representations. Many of the ideas used in the proof of these conjectures were then carried over to the Lie group setting by Lusztig-Vogan.
My own work on highest weight representations began with the papers [8,9], where a connection between highest weight representations and Lie group representations was developed and used to prove the existence of a collection of differential operators between vector bundles. The work in [10] looks at the Hermitian symmetric case and proves a multiplicity one theorem As noted above, these highest weight representations are famous, going back to Harish-Chandra's early work on the discrete portion of the Plancherel measure for a reductive Lie group. The work in [17] goes on to describe more thoroughly the structure of these highest weight representations, playing off the techniques in the paper [16] quoted above. The natural maps between highest weight representations give rise to differential operators between vector bundles. The paper [11] was the beginning of a series which ultimately classified all such operators. Finally, [15] describes how to formulate and prove a Kazhdan-Lusztig conjecture in a more general setting.
Click on picture to find out it's significance.
Click on picture to find out significance.
My most recent work in [28] attempts to bring together the theory of asymptotics, singularity theory (encoded by a certain variety attached to each representation) and generalizations of classical Whittaker functions. The underlying techniques are analytic, algebraic and geometric. To each irreducible representation of interest, we can naturally associate a variety in g* (the dual of the Lie algebra), which is the closure of a single nilpotent orbit. My book [27], coauthored with my colleague Monty McGovern, lays out the theory of nilpotent orbits on a level accessible to interested graduate students. In fact, this book grew out of a two quarter course I taught at the University of Washington with my coauthor Monty McGovern.