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

Kim, Sungwoon; Kim, Inkang
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
Tipo: Artigo de Revista Científica
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.
Tipo: Artigo de Revista Científica
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
Tipo: Artigo de Revista Científica
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
Tipo: Artigo de Revista Científica
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
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
Tipo: Artigo de Revista Científica
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
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.
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
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
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