Reductive Lie Group Research Articles

Homotopy principles for equivariant isomorphisms

Let G G be a reductive complex Lie group acting holomorphically on Stein manifolds X X and Y Y . Let p X : X → Q X p_X\colon X\to Q_X and p Y : Y → Q Y p_Y\colon Y\to Q_Y be the quotient mappings. When is there an equivariant biholomorphism of X X and Y Y ? A necessary condition is that the categorical quotients Q X Q_X and Q Y Q_Y are biholomorphic and that the biholomorphism φ \varphi sends the Luna strata of Q X Q_X isomorphically onto the corresponding Luna strata of Q Y Q_Y . Fix φ \varphi . We demonstrate two homotopy principles in this situation. The first result says that if there is a G G -diffeomorphism Φ : X → Y \Phi \colon X\to Y , inducing φ \varphi , which is G G -biholomorphic on the reduced fibres of the quotient mappings, then Φ \Phi is homotopic, through G G -diffeomorphisms satisfying the same conditions, to a G G -equivariant biholomorphism from X X to Y Y . The second result roughly says that if we have a G G -homeomorphism Φ : X → Y \Phi \colon X\to Y which induces a continuous family of G G -equivariant biholomorphisms of the fibres p X − 1 ( q ) p_X{^{-1}}(q) and p Y − 1 ( φ ( q ) ) p_Y{^{-1}}(\varphi (q)) for q ∈ Q X q\in Q_X and if X X satisfies an auxiliary property (which holds for most X X ), then Φ \Phi is homotopic, through G G -homeomorphisms satisfying the same conditions, to a G G -equivariant biholomorphism from X X to Y Y . Our results improve upon those of our earlier paper [J. Reine Angew. Math. 706 (2015), 193–214] and use new ideas and techniques.

Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics

Several problems in number theory when reformulated in terms of homogenous dynamics involve study of limiting distributions of translates of algebraically defined measures on orbits of reductive groups. The general non-divergence and linearization techniques, in view of Ratner’s measure classification for unipotent flows, reduce such problems to dynamical questions about linear actions of reductive groups on finite-dimensional vector spaces. This article provides general results which resolve these linear dynamical questions in terms of natural group theoretic or geometric conditions.

Stability with respect to actions of real reductive Lie groups

We give a systematic treatment of the stability theory for action of a real reductive Lie group G on a topological space. More precisely, we introduce an abstract setting for actions of noncompact real reductive Lie groups on topological spaces that admit functions similar to the Kempf–Ness function. The point of this construction is that one can characterize stability, semi-stability and polystability of a point by numerical criteria, that is in terms of a function called maximal weight. We apply this setting to the actions of a real noncompact reductive Lie group G on a real compact submanifold M of a Kahler manifold Z and to the action of G on measures of M.

Solvability, Structure, and Analysis for Minimal Parabolic Subgroups

We examine the structure of the Levi component MA in a minimal parabolic subgroup $$P = MAN$$ of a real reductive Lie group G and work out the cases where M is metabelian, equivalently where $$\mathfrak {p}$$ is solvable. When G is a linear group we verify that $$\mathfrak {p}$$ is solvable if and only if M is commutative. In the general case M is abelian modulo the center $$Z_G$$ , we indicate the exact structure of M and P, and we work out the precise Plancherel Theorem and Fourier Inversion Formulae. This lays the groundwork for comparing tempered representations of G with those induced from generic representations of P.

Homotopy groups of free group character varieties

Let G be a connected, complex reductive Lie group with maximal compact subgroup K, and let X denote the moduli space of G- or K-valued representations of a rank r free group. In this article, we develop methods for studying the low-dimensional homotopy groups of these spaces and of their subspaces Y of irreducible representations. Our main result is that when G = GL(n,C) or SL(n,C), the second homotopy group of X is trivial. The proof depends on a new general position-type result in a singular setting. This result is proven in the Appendix and may be of independent interest. We also obtain new information regarding the homotopy groups of the subspaces Y. Recent work of Biswas and Lawton determined the fundamental group of X for general G, and we describe the fundamental group of Y. Specializing to the case G = GL(n,C), we explicitly compute the homotopy groups of the smooth locus of X in a large range of dimensions, finding that they exhibit Bott Periodicity. As a further application of our methods (and in particular our general position result) we obtain new results regarding centralizers of subgroups of G and K, motivated by a question of Sikora. Additionally, we use work of Richardson to solve a conjecture of Florentino-Lawton about the singular locus of X, and we give a topological proof that for G= GL(n,C) or SL(n,C), the space X is not a rational Poincar\'e Duality Space for r>3 and n=2.

A Note on the Classification of Compact Homogeneous Locally Conformal Kähler Manifolds

In this study, we apply a result of H. C. Wang and Hano-Kobayashi on the classification of compact complex homogeneous manifolds with a compact reductive Lie group to give some more homogeneous space involved proofs of recent classification of compact complex homogeneous locally conformal Kahler manifolds. In particular, we prove that the semisimple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semisimple Lie group S. We also prove that as an one dimensional complex torus bundle, the metrics on the manifold is completely determined by the metrics (which is the same as the Kahler class) on the base complex manifold and the metrics (same as the Kahler class) on the complex one dimensional torus.

Anosov representations and proper actions

We establish several characterizations of Anosov representations of word hyperbolic groups into real reductive Lie groups, in terms of a Cartan projection or Lyapunov projection of the Lie group. Using a properness criterion of Benoist and Kobayashi, we derive applications to proper actions on homogeneous spaces of reductive groups.

Flat Bundles With Complex Analytic Holonomy

Let G be a connected complex Lie group or a connected amenable Lie group. We show that any flat principal G-bundle over any finite CW-complex pulls back to a trivial G-bundle over some finite covering space of the base space if and only if the derived group of the radical of G is simply connected. In particular, if G is a connected compact Lie group or a connected complex reductive Lie group, then any flat principal G-bundle over any finite CW-complex pulls back to a trivial G-bundle over some finite covering space of the base space.

Compact homogeneous locally conformally Kähler manifolds

In this paper we show as main results two structure theorems of a compact homogeneous locally conformally Kahler (or shortly l.c.K.) manifold, a holomorphic structure theorem asserting that it has a structure of holomorphic principal fiber bundle over a flag manifold with fiber a $1$-dimensional complex torus, and a metric structure theorem asserting that it is necessarily of Vaisman type. We also discuss and determine l.c.K. reductive Lie groups and compact locally homogeneous l.c.K. manifolds of reductive Lie groups.

A Plancherel formula for L2(G∕H) for almost symmetric subgroups

In this paper we study the Plancherel formula for a new class of homogeneous spaces for real reductive Lie groups; these spaces are fibered over non-Riemannian symmetric spaces, and they exhibit a phenomenon of uniform infinite multiplicities. They also provide examples of non-tempered representations of the group appearing in the Plancherel formula. Several classes of examples are given.

Representation theory in complex rank, II

We define and study representation categories based on Deligne categories Rep(GLt), Rep(Ot), Rep(Sp2t), where t is any (non-integer) complex number. Namely, we define complex rank analogs of the parabolic category O and the representation categories of real reductive Lie groups and supergroups, affine Lie algebras, and Yangians. We develop a framework and language for studying these categories, prove basic results about them, and outline a number of directions of further research.

SUFFICIENT CONDITIONS FOR HOLOMORPHIC LINEARISATION

Let G be a reductive complex Lie group acting holomorphically on X = ℂn. The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on ℂn such that the G-action becomes linear. Equivalently, is there a G-equivariant biholomorphism Φ: X → V where V is a G-module? There is an intrinsic stratification of the categorical quotient QX, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of G. Suppose that there is a Φ as above. Then Φ induces a biholomorphism φ: QX → QV which is stratified, i.e., the stratum of QX with a given label is sent isomorphically to the stratum of QV with the same label.

Compact homogeneous spaces of reductive Lie groups and spaces close to them

We study compact homogeneous spaces of reductive Lie groups, and also some of their analogues and generalizations (quasicompact and plesiocompact homogeneous spaces of these Lie groups). We give a description of the structure of (plesio-)uniform subgroups in reductive Lie groups. The corresponding homogeneous spaces for which the stationary subgroup has an extremal dimension (close to the minimal or maximal possible one) are described. The fundamental groups of (plesio)compact homogeneous spaces of arbitrary reductive and semisimple Lie groups are characterized. Bibliography: 16 titles.

Wave front sets of reductive Lie group representations

If G is a Lie group, H⊂G is a closed subgroup, and τ is a unitary representation of H, then the authors give a sufficient condition on ξ∈ig∗ to be in the wave front set of IndHGτ. In the special case where τ is the trivial representation, this result was conjectured by Howe. If G is a real, reductive algebraic group and π is a unitary representation of G that is weakly contained in the regular representation, then the authors give a geometric description of WF(π) in terms of the direct integral decomposition of π into irreducibles. Special cases of this result were previously obtained by Kashiwara–Vergne, Howe, and Rossmann. The authors give applications to harmonic analysis problems and branching problems.

Poisson transforms for differential forms

We give a construction of a Poisson transform mapping density valued differential forms on generalized flag manifolds to differential forms on the corresponding Riemannian symmetric spaces, which can be described entirely in terms of finite dimensional representations of reductive Lie groups. Moreover, we will explicitly generate a family of degree-preserving Poisson transforms whose restriction to real valued differential forms has coclosed images. In addition, as a transform on sections of density bundles it can be related to the classical Poisson transform, proving that we produced a natural generalization of the classical theory.

Ratio coordinates for higher Teichmüller spaces

We define new coordinates for Fock-Goncharov's higher Teichm\"uller spaces for a surface with holes, which are the moduli spaces of representations of the fundamental group into a reductive Lie group $G$. Some additional data on the boundary leads to two closely related moduli spaces, the $\mathscr{X}$-space and the $\mathscr{A}$-space, forming a cluster ensemble. Fock and Goncharov gave nice descriptions of the coordinates of these spaces in the cases of $G = PGL_m$ and $G=SL_m$, together with Poisson structures. We consider new coordinates for higher Teichm\"uller spaces given as ratios of the coordinates of the $\mathscr{A}$-space for $G=SL_m$, which are generalizations of Kashaev's ratio coordinates in the case $m=2$. Using Kashaev's quantization for $m=2$, we suggest a quantization of the system of these new ratio coordinates, which may lead to a new family of projective representations of mapping class groups. These ratio coordinates depend on the choice of an ideal triangulation decorated with a distinguished corner at each triangle, and the key point of the quantization is to guarantee certain consistency under a change of such choices. We prove this consistency for $m=3$, and for completeness we also give a full proof of the presentation of Kashaev's groupoid of decorated ideal triangulations.

On boundary value problems for some conformally invariant differential operators

ABSTRACTWe study boundary value problems for some differential operators on Euclidean space and the Heisenberg group which are invariant under the conformal group of a Euclidean subspace, respectively, Heisenberg subgroup. These operators are shown to be self-adjoint in certain Sobolev type spaces and the related boundary value problems are proven to have unique solutions in these spaces. We further find the corresponding Poisson transforms explicitly in terms of their integral kernels and show that they are isometric between Sobolev spaces and extend to bounded operators between certain Lp-spaces.The conformal invariance of the differential operators allows us to apply unitary representation theory of reductive Lie groups, in particular recently developed methods for restriction problems.

The Borel–Weil theorem for reductive Lie groups

In this manuscript we consider the extent to which an irreducible representation for a reductive Lie group can be realized as the sheaf cohomolgy of an equivariant holomorphic line bundle defined on an open invariant submanifold of a complex flag space. Our main result is the following: suppose $G_{0}$ is a real reductive group of Harish-Chandra class and let $X$ be the associated full complex flag space. Suppose $\mathcal{O}_{\lambda}$ is the sheaf of sections of a $G_{0}$-equivariant holomorphic line bundle on $X$ whose parameter $\lambda$ (in the usual twisted $\mathcal{D}% $-module context) is antidominant and regular. Let $S\subseteq X$ be a $G_{0}% $-orbit and suppose $U\supseteq S$ is the smallest $G_{0}$-invariant open submanifold of $X$ that contains $S$. From the analytic localization theory of Hecht and Taylor one knows that there is a nonegative integer $q$ such that the compactly supported sheaf cohomology groups $H_{\text{c}}^{q}(S,\mathcal{O}_{\lambda}\mid_{S})$ vanish except in degree $q$, in which case $H_{\text{c}}^{q}(S,\mathcal{O}_{\lambda}\mid_{S})$ is the minimal globalization of an associated standard Beilinson-Bernstein module. In this study we show that the $q$-th compactly supported cohomolgy group $H_{\text{c}}^{q}(U,\mathcal{O}_{\lambda}\mid_{U})$ defines, in a natural way, a nonzero submodule of $H_{\text{c}}^{q}(S,\mathcal{O}_{\lambda}\mid_{S})$, which is irreducible (i.e. realizes the unique irreducible submodule of $H_{\text{c}}^{q}(S,\mathcal{O}_{\lambda}\mid_{S})$) when an associated algebraic variety is nonsingular. By a tensoring argument, we can show that the result holds, more generally (for nonsingular Schubert variety), when the representation $H_{\text{c}}^{q}(S,\mathcal{O}_{\lambda}\mid_{S})$ is what we call a classifying module.

Homogeneous locally conformally Kähler and Sasaki manifolds

We prove various classification results for homogeneous locally conformally symplectic manifolds. In particular, we show that a homogeneous locally conformally Kähler manifold of a reductive group is of Vaisman type if the normalizer of the isotropy group is compact. We also show that such a result does not hold in the case of non-compact normalizer and determine all left-invariant lcK structures on reductive Lie groups.

A restriction on proper group actions on homogeneous spaces of reductive type

Let $$L$$ be a reductive subgroup of a reductive Lie group $$G$$ . Let $$G/H$$ be a homogeneous space of reductive type. We provide a necessary condition for the properness of the action of $$L$$ on $$G/H$$ . As an application we give examples of spaces that do not admit standard compact Clifford-Klein forms.