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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21 Citations
Highly Influential Citations
3
Background Citations
8
Methods Citations
3

21 Citations

Analytic torsion, dynamical zeta functions, and the Fried conjecture
We prove the equality of the analytic torsion and the value at zero of a Ruelle dynamical zeta function associated with an acyclic unitarily flat vector bundle on a closed locally symmetric reductive…
Topological Frobenius reciprocity and invariant Hermitian forms.
In his article "Unitary Representations and Complex Analysis", David Vogan gives a characterization of the continuous invariant Hermitian forms defined on the compactly supported sheaf cohomology…
On the unitary globalization of cohomologically induced modules
Complex of twistor operators in symplectic spin geometry
For a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure), we construct a sequence consisting of differential operators using a symplectic…
Symplectic Dirac Operator and its Higher Spin Analogues
Symplectic Dirac operator is a symplectic counterpart of the classical Dirac operator known from Clifford analysis or Riemannian geometry. We introduce this operator and mention some of its…
Cohomological Induction on Generalized G-Modules to Infinite Dimensional Representations
Extensions and globalizations of Harish-Chandra modules are used to obtain a representation theory that includes the cases of non-compact type of irreducible representations studied in the Vogan…
The algebraic version of a conjecture by Vogan
In a recent manuscript, D.Vogan conjectures that four canonical globalizations of Harish-Chandra modules commute with certain n-cohomology groups. In this article we focus on the case of a complex…
Classification of 1st order symplectic spinor operators over contact projective geometries
Relation of the spectra of symplectic Rarita-Schwinger and Dirac operators on flat symplectic manifolds
Consider a flat symplectic manifold (M, ω), l ≥ 2, admitting a metaplectic structure. We prove that the symplectic twistor operator maps the eigenvectors of the symplectic Dirac operator, that are…
THE n-HOMOLOGY OF REPRESENTATIONS
The n-homology groups of a g-module provide a natural and fruitful extension of the concept of highest weight to the representation theory of a noncompact reductive Lie group. In this article we give…
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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…
The Mathematical Heritage of Hermann Weyl
On induced representations by R. Bott Differentiable structures on fractal-like sets, determined by intrinsic scaling functions on dual Cantor sets by D. Sullivan Representation theory and arithmetic…
REPRESENTATION THEORY OF SEMISIMPLE GROUPS: An Overview Based on Examples
Page 55, proof of Lemma 3.13. This proof is incorrect as it stands because it involves an interchange of limits that has not been justified. A naive attempt to fix the proof might involve assuming…
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