Representation Theory and Harmonic Analysis on Semisimple Lie Groups
Paul J. Sally (Jr.), David A. Vogan
American Mathematical Soc., 1989 - Mathematics - 350 pages
This book brings together five papers that have been influential in the study of Lie groups. Though published more than 20 years ago, these papers made fundamental contributions that deserve much broader exposure. In addition, the subsequent literature that has subsumed these papers cannot replace the originality and vitality they contain. The editors have provided a brief introduction to each paper, as well as a synopsis of the major developments which have occurred in the area covered by each paper. Included here are the doctoral theses of Arthur, Osborne, and Schmid. Arthur's thesis is closely related to Trombi's paper insofar as both deal with harmonic analysis on real semisimple Lie groups, and, in particular, analysis on the Schwartz space of Harish-Chandra. Arthur's thesis is concerned with the image under the Fourier transform of the Schwartz space of a semisimple Lie group of real rank one, while Trombi's paper provides an expository account of the harmonic analysis associated to the decomposition of the Schwartz space under the regular representation. In his thesis, Osborne extends the Atiyah-Bott fixed point theorem for elliptic complexes to obtain a fixed point formula for complexes that are not elliptic. Schmid proves a generalization of the Borel-Weil theorem concerning an explicit and geometric realization of the irreducible representations of a compact, connected semisimple Lie group. Langlands's fundamental paper provides a classification of irreducible, admissible representations of real reductive Lie groups.
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On the classification of irreducible representations of real algebraic groups
Lefschetz formulas on nonelliptic complexes
Homogeneous complex manifolds and representations of semisimple Liegroups
On HarishChandras theory of the Eisenstein integral for real semisimple Lie groups
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adjoint automorphism Cartan subalgebra Cartan subgroup character choose cohomology commutes compact complex contains corollary corresponding decomposition defined over F denote discrete series element equal exists expt exptHo finite finite-dimensional fixed point follows formula Fréchet space G-invariant Harish-Chandra harmonic analysis Hence Hermitian metric Hilbert space holomorphic homogeneous infinitesimally equivalent inner product invariant irreducible representation isomorphism K-finite K-module Langlands Lemma Lie algebra line bundle linear functional Math orthogonal orthonormal parabolic subgroup polynomials positive roots principal series representations prove quasi-simple representation of G respectively restriction semisimple Lie groups ſº square-integrable subgroup of G subspace Suppose Theorem topology trace class unitary representation vanishes vector bundle vector space Weyl chamber Weyl group zero