# 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?

## 21 Citations

Analytic torsion, dynamical zeta functions, and the Fried conjecture

- Mathematics
- 2016

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.

- Mathematics
- 2020

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…

Complex of twistor operators in symplectic spin geometry

- Mathematics
- 2009

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

- Mathematics
- 2008

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

- Mathematics
- 2013

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

- Mathematics
- 2006

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

- Mathematics
- 2008

Relation of the spectra of symplectic Rarita-Schwinger and Dirac operators on flat symplectic manifolds

- Mathematics
- 2007

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

- Mathematics
- 2008

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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