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2014 | Book

Group Gradings on Lie Algebras and Applications to Geometry: II Read chapter Read first chapter

Developments and Retrospectives in Lie Theory

Geometric and Analytic Methods

Editors: Geoffrey Mason, Ivan Penkov, Joseph A. Wolf

Publisher: Springer International Publishing

Book Series : Developments in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

The Lie Theory Workshop, founded by Joe Wolf (UC, Berkeley), has been running for over two decades. At the beginning, the top universities in California and Utah hosted the meetings, which continue to run on a quarterly basis. Experts in representation theory/Lie theory from various parts of the US, Europe, Asia (China, Japan, Singapore, Russia), Canada, and South and Central America were routinely invited to give talks at these meetings. Nowadays, the workshops are also hosted at universities in Louisiana, Virginia, and Oklahoma. These Lie theory workshops have been sponsored by the NSF, noting the talks have been seminal in describing new perspectives in the field covering broad areas of current research. The contributors have all participated in these Lie theory workshops and include in this volume expository articles which will cover representation theory from the algebraic, geometric, analytic, and topological perspectives with also important connections to math physics. These survey articles, review and update the prominent seminal series of workshops in representation/Lie theory mentioned above, and reflects the widespread influence of those workshops in such areas as harmonic analysis, representation theory, differential geometry, algebraic geometry, number theory, and mathematical physics. Many of the contributors have had prominent roles in both the classical and modern developments of Lie theory and its applications.

Table of Contents

Frontmatter
Group Gradings on Lie Algebras and Applications to Geometry: II
Abstract
This paper is devoted to some applications of the theory of group gradings on Lie algebras to two topics of differential geometry, such as generalized symmetric manifolds and affine structures on nilmanifolds.
Yuri Bahturin, Michel Goze, Elisabeth Remm
Harmonic Analysis on Homogeneous Complex Bounded Domains and Noncommutative Geometry
Abstract
We define and study a noncommutative Fourier transform on every homogeneous complex bounded domain. We then give an application in noncommutative differential geometry by defining noncommutative Baumslag–Solitar tori.
Pierre Bieliavsky, Victor Gayral, Axel de Goursac, Florian Spinnler
The Radon Transform and Its Dual for Limits of Symmetric Spaces
Abstract
The Radon transform and its dual are central objects in geometric analysis on Riemannian symmetric spaces of the noncompact type. In this article we study algebraic versions of those transforms on inductive limits of symmetric spaces. In particular, we show that normalized versions exists on some spaces of regular functions on the limit. We give a formula for the normalized transform using integral kernels and relate them to limits of double fibration transforms on spheres.
Joachim Hilgert, Gestur Ólafsson
Cycle Connectivity and Automorphism Groups of Flag Domains
Abstract
A flag domain D is an open orbit of a real form G 0 in a flag manifold \(Z = G/P\) of its complexification. If D is holomorphically convex, then, since it is a product of a Hermitian symmetric space of bounded type and a compact flag manifold, Aut(D) is easily described. If D is not holomorphically convex, then in previous work it was shown that Aut(D) is a Lie group whose connected component at the identity agrees with G 0, except possibly in situations which arise in Onishchik’s list of flag manifolds where \(\mathrm{Aut}(Z)^{0} =\hat{ G}\) is larger than G. In the present work the group \(\mathrm{Aut}(D)^{0} =\hat{ G}_{0}\) is described as a real form of \(\hat{G}\). Using an observation of Kollar, new and much simpler proofs of much of our previous work in the case where D is not holomorphically convex are given.
Alan Huckleberry
Shintani Functions, Real Spherical Manifolds, and Symmetry Breaking Operators
Abstract
For a pair of reductive groups G ⊃ G′, we prove a geometric criterion for the space Sh(λ, ν) of Shintani functions to be finite-dimensional in the Archimedean case. This criterion leads us to a complete classification of the symmetric pairs (G, G′) having finite-dimensional Shintani spaces. A geometric criterion for uniform boundedness of \(\dim _{\mathbb{C}}\mathrm{Sh}(\lambda,\nu )\) is also obtained. Furthermore, we prove that symmetry breaking operators of the restriction of smooth admissible representations yield Shintani functions of moderate growth, of which the dimension is determined for \((G,G') = (O(n + 1,1),O(n,1))\).
Toshiyuki Kobayashi
Harmonic Spinors on Reductive Homogeneous Spaces
Abstract
An integral intertwining operator is given from certain principal series representations into spaces of harmonic spinors for Kostant’s cubic Dirac operator. This provides an integral representation for harmonic spinors on a large family of reductive homogeneous spaces.
Salah Mehdi, Roger Zierau
Twisted Harish–Chandra Sheaves and Whittaker Modules: The Nondegenerate Case
Abstract
In this paper we develop a geometric approach to the study of the category of Whittaker modules. As an application, we reprove a well-known result of B. Kostant on the structure of the category of nondegenerate Whittaker modules.
Dragan Miličić, Wolfgang Soergel
Unitary Representations of Unitary Groups
Abstract
In this paper we review and streamline some results of Kirillov, Olshanski and Pickrell on unitary representations of the unitary group \(\mathop{\mathrm{U}}\nolimits (\mathcal{H})\) of a real, complex or quaternionic separable Hilbert space and the subgroup \(\mathop{\mathrm{U}}\nolimits _{\infty }(\mathcal{H})\), consisting of those unitary operators g for which g −1 is compact. The Kirillov–Olshanski theorem on the continuous unitary representations of the identity component \(\mathop{\mathrm{U}}\nolimits _{\infty }(\mathcal{H})_{0}\) asserts that they are direct sums of irreducible ones which can be realized in finite tensor products of a suitable complex Hilbert space. This is proved and generalized to inseparable spaces. These results are carried over to the full unitary group by Pickrell’s theorem, asserting that the separable unitary representations of \(\mathop{\mathrm{U}}\nolimits (\mathcal{H})\), for a separable Hilbert space \(\mathcal{H}\), are uniquely determined by their restriction to \(\mathop{\mathrm{U}}\nolimits _{\infty }(\mathcal{H})_{0}\). For the 10 classical infinite rank symmetric pairs (G, K) of non-unitary type, such as \((\mathop{\mathrm{GL}}\nolimits (\mathcal{H}),\mathop{\mathrm{U}}\nolimits (\mathcal{H}))\), we also show that all separable unitary representations are trivial.
Karl-Hermann Neeb
Weak Splittings of Quotients of Drinfeld and Heisenberg Doubles
Abstract
We investigate the fine structure of the symplectic foliations of Poisson homogeneous spaces. Two general results are proved for weak splittings of surjective Poisson submersions from Heisenberg and Drinfeld doubles. The implications of these results are that the torus orbits of symplectic leaves of the quotients can be explicitly realized as Poisson–Dirac submanifolds of the torus orbits of the doubles. The results have a wide range of applications to many families of real and complex Poisson structures on flag varieties. Their torus orbits of leaves recover important families of varieties such as the open Richardson varieties.
Milen Yakimov
Metadata
Title
Developments and Retrospectives in Lie Theory
Editors
Geoffrey Mason
Ivan Penkov
Joseph A. Wolf
Copyright Year
2014
Electronic ISBN
978-3-319-09934-7
Print ISBN
978-3-319-09933-0
DOI
https://doi.org/10.1007/978-3-319-09934-7

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