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Volume invariant and maximal representations of discrete subgroups of Lie groups

Kim, Sungwoon; Kim, Inkang
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.38%
Let $\Gamma$ be a lattice in a connected semisimple Lie group $G$ with trivial center and no compact factors. We introduce a volume invariant for representations of $\Gamma$ into $G$, which generalizes the volume invariant for representations of uniform lattices introduced by Goldman. Then, we show that the maximality of this volume invariant exactly characterizes discrete, faithful representations of $\Gamma$ into $G$ except for $\Gamma\subset \mathrm{PSL_2 \mathbb{C}}$ a nonuniform lattice.; Comment: 28 pages

Uniform oscillatory behavior of spherical functions of $GL_n/U_n$ at the identity and a central limit theorem

Voit, Michael
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 22/05/2012
Relevância na Pesquisa
16.38%
Let $\mathbb F=\mathbb R$ or $\mathbb C$ and $n\in\b N$. Let $(S_k)_{k\ge0}$ be a time-homogeneous random walk on $GL_n(\b F)$ associated with an $U_n(\b F)$-biinvariant measure $\nu\in M^1(GL_n(\b F))$. We derive a central limit theorem for the ordered singular spectrum $\sigma_{sing}(S_k)$ with a normal distribution as limit with explicit analytic formulas for the drift vector and the covariance matrix. The main ingredient for the proof will be a oscillatory result for the spherical functions $\phi_{i\rho+\lambda}$ of $(GL_n(\b F),U_n(\b F))$. More precisely, we present a necessarily unique mapping $m_{\bf 1}:G\to\b R^n$ such that for some constant $C$ and all $g\in G$, $\lambda\in\b R^n$, $$|\phi_{i\rho+\lambda}(g)- e^{i\lambda\cdot m_{\bf 1}(g)}|\le C\|\lambda\|^2.$$

The Paley-Wiener Theorem and Limits of Symmetric Spaces

Olafsson, Gestur; Wolf, Joseph A.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 23/01/2011
Relevância na Pesquisa
16.38%
We extend the Paley--Wiener theorem for riemannian symmetric spaces to an important class of infinite dimensional symmetric spaces. For this we define a notion of propagation of symmetric spaces and examine the direct (injective) limit symmetric spaces defined by propagation. This relies on some of our earlier work on invariant differential operators and the action of Weyl group invariant polynomials under restriction.

Detecting orbits along subvarieties via the moment map

Jablonski, Michael
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 31/10/2008
Relevância na Pesquisa
16.38%
Let G be a (real or complex) linear reductive algebraic group acting on an affine variety V. Let W be a subvariety. In this work we study how the G-orbits intersect W. We develop a criterion to determine when the intersection can be described as a finite union of orbits of a reductive subgroup. The conditions of the criterion are easily verified in practice and are used to construct continuous families of (non-isomorphic) nilpotent Lie groups which do not admit left-invariant Ricci soliton metrics. Other applications to the left-invariant geometry of Lie groups are also given. The note finishes by applying our techniques to the adjoint representation. The classical result of finiteness of nilpotent orbits is reproven and it is shown that each of these orbits contains a critical point of the norm squared of the moment map.; Comment: 15 pages

On the limit set of Anosov representations

Kim, Inkang; Kim, Sungwoon
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 04/12/2012
Relevância na Pesquisa
16.38%
We study the limit set of discrete subgroups arising from Anosov representations. Specially we study the limit set of discrete groups arising from strictly convex real projective structures and Anosov representations from a finitely generated word hyperbolic group into a semisimple Lie group.; Comment: 25 pages

The Hodge theory of Soergel bimodules

Elias, Ben; Williamson, Geordie
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.38%
We prove Soergel's conjecture on the characters of indecomposable Soergel bimodules. We deduce that Kazhdan-Lusztig polynomials have positive coefficients for arbitrary Coxeter systems. Using results of Soergel one may deduce an algebraic proof of the Kazhdan-Lusztig conjecture.; Comment: 44 pages. v2: many minor changes, final version

Attractor Networks on Complex Flag Manifolds

Hilgert, Joachim
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 13/12/2008
Relevância na Pesquisa
16.38%
Robbin and Salamon showed that attractor-repellor networks and Lyapunov maps are equivalent concepts and illustrate this with the example of linear flows on projective spaces. In these examples the fixed points are linearly ordered with respect to the Smale order which makes the attractor-repellor network overly simple. In this paper we provide a class of examples in which the attractor-repellor network and its lattice structure can be explicitly determined even though the Smale order is not total. They are associated with special flows on complex flag manifolds. In the process we show that the Smale order on the set of fixed points can be identified with the well-known Bruhat order. This could also be derived from results of Kazhdan and Lusztig, but we give a new proof using the Lambda-Lemma of Palis. For the convenience of the reader we also introduce the flag manifolds via elementary dynamical systems using only a minimum of Lie theory.

On the restriction of Zuckerman's derived functor modules A_q(\lambda) to reductive subgroups

Oshima, Yoshiki
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.38%
In this article, we study the restriction of Zuckerman's derived functor (g,K)-modules A_q(\lambda) to g' for symmetric pairs of reductive Lie algebras (g,g'). When the restriction decomposes into irreducible (g',K')-modules, we give an upper bound for the branching law. In particular, we prove that each (g',K')-module occurring in the restriction is isomorphic to a submodule of A_q'(\lambda') for a parabolic subalgebra q' of g', and determine their associated varieties. For the proof, we construct A_q(\lambda) on complex partial flag varieties by using D-modules.; Comment: 30 pages, typos corrected, references added

Linear maps preserving invariants

Schwarz, Gerald W.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.38%
Let $G\subset\GL(V)$ be a complex reductive group. Let $G'$ denote $\{\phi\in\GL(V)\mid p\circ\phi=p\text{for all} p\in\C[V]^G\}$. We show that, in general, $G'=G$. In case $G$ is the adjoint group of a simple Lie algebra $\lieg$, we show that $G'$ is an order 2 extension of $G$. We also calculate $G'$ for all representations of $\SL_2$.; Comment: minor changes

Microlocal analysis on wonderful varieties. Regularized traces and global characters

Cupit-Foutou, Stephanie; Parthasarathy, Aprameyan; Ramacher, Pablo
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.38%
Let $\mathbf{G}$ be a connected reductive complex algebraic group with split real form $(G,\sigma)$. Consider a strict wonderful $\mathbf{G}$-variety $\bf{X}$ equipped with its $\sigma$-equivariant real structure, and let $X$ be the corresponding real locus. Further, let $E$ be a real differentiable $G$-vector bundle over $X$. In this paper, we introduce a distribution character for the regular representation of $G$ on the space of smooth sections of $E$, and show that on a certain open subset of $G$ of transversal elements it is locally integrable and given by a sum over fixed points.; Comment: Revised version, 22 pages

Zhu's algebras, $C_2$-algebras and abelian radicals

Feigin, Boris; Feigin, Evgeny; Littelmann, Peter
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.38%
This paper consists of three parts. In the first part we prove that Zhu's and $C_2$-algebras in type $A$ have the same dimensions. In the second part we compute the graded decomposition of the $C_2$-algebras in type $A$, thus proving the Gaberdiel-Gannon's conjecture. Our main tool is the theory of abelian radicals, which we develop in the third part.; Comment: 19 pages

Quantisation of presymplectic manifolds, K-theory and group representations

Hochs, Peter
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.38%
Let $G$ be a semisimple Lie group with finite component group, and let $K

McKay correspondence and the branching law for finite subgroups of $\mathbf{SL}_3\mathbb{C}$

Butin, Frédéric; Perets, Gadi S.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 03/09/2009
Relevância na Pesquisa
16.38%
Given $\Gamma$ a finite subgroup of $\mathbf{SL}_3\mathbb{C}$, we determine how an arbitrary finite dimensional irreducible representation of $\mathbf{SL}_3\mathbb{C}$ decomposes under the action of $\Gamma$. To the subgroup $\Gamma$ we attach a generalized Cartan matrix $C_\Gamma$. Then, inspired by B. Kostant, we decompose the Coxeter element of the Kac-Moody algebra attached to $C_\Gamma$ as a product of reflections of a special form, thereby suggesting an algebraic form for the McKay correspondence in dimension 3.; Comment: 32 pages

Curved Casimir Operators and the BGG Machinery

Cap, Andreas; Soucek, Vladimir
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.38%
We prove that the Casimir operator acting on sections of a homogeneous vector bundle over a generalized flag manifold naturally extends to an invariant differential operator on arbitrary parabolic geometries. We study some properties of the resulting invariant operators and compute their action on various special types of natural bundles. As a first application, we give a very general construction of splitting operators for parabolic geometries. Then we discuss the curved Casimir operators on differential forms with values in a tractor bundle, which nicely relates to the machinery of BGG sequences. This also gives a nice interpretation of the resolution of a finite dimensional representation by (spaces of smooth vectors in) principal series representations provided by a BGG sequence.; Comment: This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

A simple proof of the algebraic version of a conjecture by Vogan

Bratten, Tim; Corti, Sergio
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.38%
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 prove that Vogan's conjecture holds for one of the globalizations if and only if it holds for the dual. Using a previously published result of one of the authors, which establishes the conjecture for the minimal globalization, we can therefore deduce Vogan's conjecture for the maximal globalization.; Comment: A simple proof of the problem raised in arXiv:math/0606071v1

A Method for Weight Multiplicity Computation Based on Berezin Quantization

Bar-Moshe, David
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.38%
Let $G$ be a compact semisimple Lie group and $T$ be a maximal torus of $G$. We describe a method for weight multiplicity computation in unitary irreducible representations of $G$, based on the theory of Berezin quantization on $G/T$. Let $\Gamma_{\rm hol}(\mathcal{L}^{\lambda})$ be the reproducing kernel Hilbert space of holomorphic sections of the homogeneous line bundle $\mathcal{L}^{\lambda}$ over $G/T$ associated with the highest weight $\lambda$ of the irreducible representation $\pi_{\lambda}$ of $G$. The multiplicity of a weight $m$ in $\pi_{\lambda}$ is computed from functional analytical structure of the Berezin symbol of the projector in $\Gamma_{\rm hol}(\mathcal{L}^{\lambda})$ onto subspace of weight $m$. We describe a method of the construction of this symbol and the evaluation of the weight multiplicity as a rank of a Hermitian form. The application of this method is described in a number of examples.

On the kernel of the maximal flat Radon transform on symmetric spaces of compact type

Grinberg, Eric L.; Jackson, Steven Glenn
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 27/06/2014
Relevância na Pesquisa
16.38%
Let $M$ be a Riemannian globally symmetric space of compact type, $M'$ its set of maximal flat totally geodesic tori, and $\mathrm{ad}(M)$ its adjoint space. We show that the kernel of the maximal flat Radon transform $\tau:L^2(M) \rightarrow L^2(M')$ is precisely the orthogonal complement of the image of the pullback map $L^2(\mathrm{ad}(M))\rightarrow L^2(M)$. In particular, we show that the maximal flat Radon transform is injective if and only if $M$ coincides with its adjoint space.; Comment: 10 pages

Complexe canonique de deuxi\`{e}me esp\`{e}ce, vari\'{e}t\'{e} commutante et bic\^{o}ne nilpotent d'une alg\`{e}bre de Lie r\'{e}ductive

Charbonnel, Jean-Yves
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 13/09/2005
Relevância na Pesquisa
16.38%
Let $g$ be a finite dimensional complex reductive Lie algebra and <.,.> an invariant non degenerated bilinear form on $g\times g$ which extends the Killing form of $[g,g]$. We define a subcomplex $E\_{\bullet}(g)$ of the canonical complex $C\_{\bullet}(g)$ of $g$. There exists a well defined sub-module $B\_{g}$ of the module of polynomial maps from $g\times g$ to $g$ which is free of rank equal to the dimension b of the borel subalgebras of $g$. Moreover, $B\_{g}$ is contained in the space of cycles of the canonical complex of $g$. The complex $E\_{\bullet}(g)$ is the ideal of $C\_{\bullet}(g)$ generated the exterior power of degree b of the module $B\_{g}$. We denote by ${\cal N}\_{g}$ the set of elements in $g\times g$ whose components generate a subsbspace contained in the nilpotent cone of $g$ and we say that $g$ has property (N) if the codimension of ${\cal N}\_{g}$ in $g\times g$ is strictly bigger than the dimension of the space of nilpotent elements in a borel subalgebra of $g$. Let $I\_{g}$ be the ideal of polynomial functions on $g\times g$ generated by the functions whose value in $(x,y)$ is the scalar product of $v$ and $[x,y]$ where $v$ is in $g$. The main result is the theorem: Let us suppose that for any semi-simple element in $g$...

Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions

Fourier, Ghislain; Littelmann, Peter
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.38%
We study finite dimensional representations of current algebras, loop algebras and their quantized versions. For the current algebra of a simple Lie algebra of type {\tt ADE}, we show that Kirillov-Reshetikhin modules and Weyl modules are in fact all Demazure modules. As a consequence one obtains an elementary proof of the dimension formula for Weyl modules for the current and the loop algebra. Further, we show that the crystals of the Weyl and the Demazure module are the same up to some additional label zero arrows for the Weyl module. For the current algebra $\Lgc$ of an arbitrary simple Lie algebra, the fusion product of Demazure modules of the same level turns out to be again a Demazure module. As an application we construct the $\Lgc$-module structure of the Kac-Moody algebra $\Lhg$-module $V(\ell\Lam_0)$ as a semi-infinite fusion product of finite dimensional $\Lgc$--modules.

Geometric representation theory for unitary groups of operator algebras

Beltiţ\ua, Daniel; Ratiu, Tudor S.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 05/01/2005
Relevância na Pesquisa
16.9%
Geometric realizations for the restrictions of GNS representations to unitary groups of $C^*$-algebras are constructed. These geometric realizations use an appropriate concept of reproducing kernels on vector bundles. To build such realizations in spaces of holomorphic sections, a class of complex coadjoint orbits of the corresponding real Banach-Lie groups are described and some homogeneous holomorphic Hermitian vector bundles that are naturally associated with the coadjoint orbits are constructed.; Comment: 17 pages