David A. Vogan

Reducibility of generalized principal series representations

(normalized induction) a generalized principal series representation. When v is unitary, these are the representations occurring in Harish-Chandras Plancherel formula for G; and for general v they may be expected to play something of the same role in harmonic analysis on G as complex characters do in R n. Langlands has shown tha t any irreducible admissible representation of G can be realized canonically as a subquotient of a generalized principal series representation (Theorem 2.9 below). For these reasons and others (some of which will be discussed below) one would like to understand the reducibility of these representations, and it is this question which motivates the results of this paper. We prove

The Langlands classification and irreducible characters for real reductive groups

1. Introduction.- 2. Structure theory: real forms.- 3. Structure theory: extended groups and Whittaker models.- 4. Structure theory: L-groups.- 5. Langlands parameters and L-homomorphisms.- 6. Geometric parameters.- 7. Complete geometric parameters and perverse sheaves.- 8. Perverse sheaves on the geometric parameter space.- 9. The Langlands classification for tori.- 10. Covering groups and projective representations.- 11. The Langlands classification without L-groups.- 12. Langlands parameters and Cartan subgroups.- 13. Pairings between Cartan subgroups and the proof of Theorem 10.4.- 14. Proof of Propositions 13.6 and 13.8.- 15. Multiplicity formulas for representations.- 16. The translation principle, the Kazhdan-Lusztig algorithm, and Theorem 1.24.- 17. Proof of Theorems 16.22 and 16.24.- 18. Strongly stable characters and Theorem 1.29.- 19. Characteristic cycles, micro-packets, and Corollary 1.32.- 20. Characteristic cycles and Harish-Chandra modules.- 21. The classification theorem and Harish-Chandra modules for the dual group.- 22. Arthur parameters.- 23. Local geometry of constructible sheaves.- 24. Microlocal geometry of perverse sheaves.- 25. A fixed point formula.- 26. Endoscopic lifting.- 27. Special unipotent representations.- References.

Unipotent representations of complex semisimple groups

algebraic group over R or C. In this paper, we restrict attention to C. We generalize Arthurs definition slightly (or perhaps simply make it more precise). All of the resulting representations, except for a finite set, are then unitarily induced from representations of the same kind on proper parabolic subgroups. We call the finite set remaining special unipotent representations; a precise definition will be given later (Definition 1.17). Our main result (Theorem III of this introduction) is a character formula for any special unipotent representation. Of course such a formula can be deduced from the Kazhdan-Lusztig conjecture (cf. [V3]). The advantages of Theorem III are that it is in closed form, and that it lends itself to verification of some conjectures of Arthur in [A]. So let G be a connected complex semisimple Lie group, and q its Lie algebra. Choose

The local structure of characters

Let G be a connected real semisimple Lie group with Lie algebra g. Let g = t + s be the Cartan decomposition and K the maximal compact subgroup with Lie algebra t. Let Θ be the character of an irreducible representation. Then Θ has an asymptotic expansion at zero (in the sense of Taylor series). As consequences of this expansion we obtain results about the asymptotic directions in which the K-types occur and about the Gelfand-Kirillov dimension of the representation.

Primitive ideals and orbital integrals in complex exceptional groups

In [2], two related problems were studied for complex classical groups: determination of the primitive spectrum of the enveloping algebra (as a set), and Fourier inversion of unipotent orbital integrals. The first of these was solved completely (extending work of Joseph and others in SL(n, C) and small groups). Although our techniques presumably solve the second as well, up to determination of normalizing constants, we carried out the calculations only for special unipotent classes (see [ 181). We had, of course, tried to apply the same methods to the exceptional groups, but they seem to be inadequate. Since that work was done, however, Brylinski and Kashiwara and Beilinson and Bernstein have established the Kazhdan-Lusztig conjecture, giving character formulas for irreducible highest weight modules [3, 71. Because of a conjecture of Joseph proved in [24], this determines in principle the primitive ideals with a fixed regular integral infinitesimal character: they are in one-to-one correspondence with what Kazhdan and Lusztig call left cells in the Weyl group. However, the algorithm given by Kazhdan and Lusztig to compute these cells is enormously complicated, requiring one to compute roughly 1 I%‘]* polynomials of degrees on the order of half the number of positive roots. This is very unsatisfactory. However, it

Functions on the model orbit in E8

First published in Representation Theory in Vol.2,1998. Published by the American Mathematical Society.

Unitary shimura correspondences for split real groups

We find a relationship between certain complementary series representations for nonlinear coverings of split simple groups, and spherical complementary series for (different) linear groups. The main technique is Barbaschs method of calculating some intertwining operators purely in terms of the Weyl group

On the classification of unitary representations of reductive Lie groups

Each subset is identified conjecturally (Conjecture 0.6) with a collection of unitary representations of a certain subgroup G(λu) of G. (We will give strong evidence and partial results for this conjecture.) In this way the problem of classifying Πu(G) would be reduced (by induction on the dimension of G) to the case G(λu) = G. Before considering the general program in more detail, we describe it in the familiar case G = SL(2,R). (This example will be treated more com-

Weyl Group Representations and Nilpotent Orbits

In [B-V2] and [B-V3] two related problems are studied for complex semisimple groups. One, is to classify the primitive ideals in the enveloping algebra. The other is to study Fourier inversion of unipotent orbital integrals.

Analysis in space-time bundles IV. Natural bundles deforming into and composed of the same invariant factors as the spin and form bundles

Abstract We study two interesting new bundles over the universal cosmos M (or maximal isotropic space-time), which may be physically applicable. The treatment is from a homogeneous vector bundle point of view and uses the notation and some of the results of the treatment in Papers I–III ( S. M. Paneitz and I. E. Segal, J. Funct. Anal. 47 (1982), 78–142; 49 (1982), 335–414; 54 (1983) , 18–22)) of conventional bundles over M. The “spannor” bundle deforms into essentially the usual spinor bundle as a conformally invariant parameter that may be interpreted as the space curvature becomes arbitrarily small. From a Minkowski space standpoint, however, the spannors involve a nontrivial action of space-time translations that deforms into a trivial action in the spinor limit and also have more complex transformation properties under discrete symmetries. Also studied are the “plyors,” consisting of the dual to the bundle product of the spannors with themselves. Composition series for the spannor and plyor section spaces are treated, relative to the conformal group, and irreducible subquotients are identified with certain that occur in conventional bundles. In particular, factors corresponding to the Maxwell and massless Dirac equations, and which may represent certain of the observed elementary particles, are determined. A gauge and conformally invariant nonlinear coupling between spannors and plyors, constituting essentially a generalization of that used in quantum electrodynamics, is developed, and an associated invariant nonlinear partial differential equation is derived. Covariant and causal quantization for spannors (as fermions) and plyors (as bosons) is formulated algebraically. The present treatment is basically mathematical, but physical motivations and possible interpretations are briefly noted.