Unitary Representations and Complex Analysis

@inproceedings{Vogan2008UnitaryRA,
  title={Unitary Representations and Complex Analysis},
  author={David A. Vogan},
  year={2008}
}
One of the great classical theorems of representation theory is the Borel-Weil Theorem, which realizes any irreducible unitary representation of a compact connected Lie group as the space of holomorphic sections of a complex line bundle on a homogeneous space. These lectures concern two closely related questions: 1. Why should something like the Borel-Weil Theorem be true? 2. Can it usefully be extended to other Lie groups? 
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References

SHOWING 1-10 OF 42 REFERENCES
Unitary Representations of Reductive Lie Groups.
This book is an expanded version of the Hermann Weyl Lectures given at the Institute for Advanced Study in January 1986. It outlines some of what is now known about irreducible unitary
A comparison theorem for Lie algebra homology groups
Let M be a Harish-Chandra module associated to a nite length, admissible representation of real reductive Lie group G0. Suppose thatp is a parabolic subalgebra of the complexied Lie algebra of G0 and
Lie Algebra Cohomology and the Generalized Borel-Weil Theorem
The present paper will be referred to as Part I. A subsequent paper entitled, “Lie algebra cohomology and generalized Schubert cells,” will be referred to as Part II.
Representations of Semisimple Lie Groups: II.
  • Harish-Chandra
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1951
TLDR
It is shown that every irreducible representation of the group (at least in case it is complex) is infinitesimally equivalent to one constructed in a certain standard way (Theorem 4).
Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups
The Description for this book, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. (AM-94), will be forthcoming.
Geometric Analysis on Symmetric Spaces
A duality in integral geometry A duality for symmetric spaces The fourier transform on a symmetric space The Radon transform on $X$ and on $X_o$. Range questions Differential equations on symmetric
Transcending classical invariant theory
Let Sp2, (R) = Sp2, = Sp be the real symplectic group of rank n. Let Sp denote the two-fold cover of Sp. There is a unitary representation, constructed by Shale [Sh] and Weil [WA], of Sp that is of
Representations of semisimple Lie groups
We keep to the notation of the preceding note.1 Since mo is reductive, there will be no essential loss of generality from the point of view of irreducible unitary representations of Mo if we assume
Canonical Extensions of Harish-Chandra Modules to Representations of G
Let G be the group of R-rational points on a reductive, Zariskiconnected, algebraic group defined over R, let K be a maximal compact subgroup, and let g be the corresponding complexified Lie algebra
Topological Vector Spaces
Preface In the notion of a topological vector space, there is a very nice interplay between the algebraic structure of a vector space and a topology on the space, basically so that the vector space
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