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## Volume invariant and maximal representations of discrete subgroups of Lie groups

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/05/2012

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#Mathematics - Classical Analysis and ODEs#Mathematics - Probability#43A90, 33C67, 22E46, 60B15, 60F05, 43A62

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 $g\in G$, $\lambda\in\b R^n$,
$$|\phi_{i\rho+\lambda}(g)- e^{i\lambda\cdot m_{\bf 1}(g)}|\le
C\|\lambda\|^2.$$

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## The Paley-Wiener Theorem and Limits of Symmetric Spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/01/2011

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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.

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## Detecting orbits along subvarieties via the moment map

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/10/2008

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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

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## On the limit set of Anosov representations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/12/2012

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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

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## The Hodge theory of Soergel bimodules

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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## Attractor Networks on Complex Flag Manifolds

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 13/12/2008

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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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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

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## Linear maps preserving invariants

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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

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## Microlocal analysis on wonderful varieties. Regularized traces and global characters

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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

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## Quantisation of presymplectic manifolds, K-theory and group representations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Symplectic Geometry#Mathematics - K-Theory and Homology#Mathematics - Representation Theory#53D50 (primary), 22E46, 46L80, 53D20 (secondary)

Let $G$ be a semisimple Lie group with finite component group, and let $K

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/09/2009

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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

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## Curved Casimir Operators and the BGG Machinery

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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/

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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## A Method for Weight Multiplicity Computation Based on Berezin Quantization

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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#Mathematical Physics#Mathematics - Representation Theory#22E46,32M05, 32M10 (Primary)#53D50, 81Q70 (Secondary)

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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/06/2014

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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

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## 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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 13/09/2005

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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 $g$...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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.

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## Geometric representation theory for unitary groups of operator algebras

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/01/2005

Relevância na Pesquisa

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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

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