Semisimple Real Group Research Articles

Character formulas from matrix factorizations

abstract: In this paper I present a new and unified method of proving character formulas for discrete series representations of connected Lie groups by applying a Chern character-type construction to the matrix factorizations of Freed-Hopkins-Teleman. In the case of a compact group I recover the Kirillov formula, thereby exhibiting the work of Freed-Teleman as a categorification of the Kirillov correspondence. In the case of a real semisimple group I recover the Rossman character formula with only a minimal amount of analysis. The appeal of this method is that it relies almost entirely on highest-weight theory, which is a far more ubiquitous phenomenon than the varied techniques that were previously used to prove such formulas.

On uniform lattices in real semisimple groups

In this article we prove that the co-compactness of the arithmetic lattices in a connected semisimple real Lie group is preserved if the lattices under consideration are representation equivalent. This is in the spirit of the question posed by Gopal Prasad and A. S. Rapinchuk in 2014 where instead of representation equivalence, the lattices under consideration are weakly commensurable Zariski dense subgroups.

A Poisson Structure on Compact Symmetric Spaces

We present some basic results on a natural Poisson structure on any compact symmetric space. The symplectic leaves of this structure are related to the orbits of the corresponding real semisimple group on the complex flag manifold.

Cohomology of buildings and of their automorphism groups

Buildings were created by Tits. They serve as a vehicle for understanding real semi-simple groups and their p-adic analogs. These are undoubtedly the two most interesting cases, but existing constructions provide many more interesting examples, some of which are remarkably symmetric. While classical (spherical or Euclidean) buildings correspond to spherical or Euclidean reflection groups, we are mainly concerned with buildings related to other (infinite) reflection groups. The class of examples which this paper addresses is that of Kac–Moody buildings, as described by Tits [Ti]. It is a very large family, and it is precisely the great symmetry of these buildings that we are using. Our initial interest in the cohomology of buildings was that we suspected that buildings provided a rich source of Kazhdan groups. Recall that a locally compact group G is a Kazhdan group (or has Property (T)) if the trivial representation of G is isolated in the space of all irreducible unitary representations of G. Equivalently, G is a Kazhdan group if H1 ct(G, ρ) = 0: i.e., its first continuous cohomology group with coefficients in any unitary representation vanishes. In this paper we use the latter definition (for their equivalence, cf. [HV]). It turns out that not only can we give an almost definitive statement about when the automorphism group of a Kac–Moody

Local Differential Geometry on the Tempered Dual of a Semisimple Lie Group

Delorme proved that the Fell topology on the tempered dual of a real semi simple group G is rather simple: roughly speaking, it is identical with the "parameter topology." The aim of this paper is to prove that the "differential geometry" of the tempered dual is very simple, too; by differential geometry, we mean three types of objects: the categories of finite length (g, K)-modules with tempered subquotients, the Extn-groups between such modules, and the deformations of such modules.

Multiplicity of the spectrum of representations connected with integrable G-invariant Hamiltonian systems

In [i, 2, 8] the case when M is a cotangent fibering to a homogeneous space G/K of a connected real semisimple group G, and the subgroup K is compact, was studied. In these papers the following theorem was proved: the integrability of the natural action of G on T*(G/K) is equivalent to the simplicity of the spectrum of the quasiregular representation of G by shifts in L=(G/K). Examples of such G D K, which are called spherical pairs or Gelfand pairs, are symmetric pairs.

Multiplicity one fails for $p$-adic unitary principal series

For a real semisimple group of matrices, every unitary principal series representation splits into inequivalent irreducible representations [5]. This multiplicity-one result was conjectured [7] in 1971 because one expected in general and knew in some special cases that all reducibility was accounted for by canonical geometric constructions, such as the imbedding of a space of analytic functions on the disc in the space of functions on the circle by passage to boundary values. We shall give an example to show that the corresponding multiplicity-one statement is false for a semisimple group of matrices defined over a locally compact, totally disconnected, nondiscrete field of characteristic 0. This example is summarized in §2, and its properties are verified in §5. It is ultimately motivated by the work of Langlands [11] on classification of irreducible admissible representations. Langlands [12] was able to give a formulation of the results of [8] that suggests that straightforward generalization of the multiplicityone theorem to other fields is not likely to succeed. An exposition of [12] is given in [9], and the way in which this work motivates our example is explained in § 6. Verification of the properties of our example depends on a suitable development of intertwining operators for split groups. Most of such a development has been carried out by Sally [15] and Winarsky [18], Some small modifications and elaborations of their work are the subject of § 3 and 4. The work in this paper grew out of conversations at the American Mathematical Society Summer Institute in 1977 with Langlands, Lusztig, Schiffmann, Shelstad, and Wallach. The work by I. Muller [13] on intertwining operators was also of influence; Muller developed a variation on Winarsky's work and was able to push through analogs of the results of [8]. In addition, she came close to discovering the example of § 2. We thank all these people* for their help. The results in this paper were announced in a talk at Hiroshima University on August 30, 1977, and later in Notices of the American Mathematical Society 25 (1978), abstract 753-G6, in connection with a meeting of the American Mathematical Society.

Minimal K-types for certain representations of real semisimple groups

Minimal K-types for certain representations of real semisimple groups

Erratum to "On the Automorphism Group of a Lie Group"

D. Z. Djokovic and G. P. Hochschild have pointed out that the argument in the third paragraph of the proof of Theorem 1 is insufficient. The fallacy lies in assuming that the fixer of S in K(S) is an algebraic subgroup. This is of course correct for complex groups, since a connected complex semisimple Lie group is algebraic, but false for real Lie groups. By Proposition 1, the theorem is correct for real solvable Lie groups and the given proof is correct for complex groups, but the theorem fails for real semisimple groups, as the following example, worked out jointly with Professor Hochschild, shows. Let G = SL(2, R); then Gc = SL(2, C) is the complexification of G and the complex conjugation T in SL(2, C) fixes exactly SL(2, R). Let

The action of a real semisimple group on a complex flag manifold. I: Orbit structure and holomorphic arc components

The action of a real semisimple group on a complex flag manifold. I: Orbit structure and holomorphic arc components