Noncompact Semisimple Lie Group Research Articles

On two geometric means and sum of adjoint orbits

In this paper, we study the metric geometric mean introduced by Pusz and Woronowicz and the spectral geometric mean introduced by Fiedler and Pták, originally for positive definite matrices. The relation between t-metric geometric mean and t-spectral geometric mean is established via log majorization. The result is then extended in the context of symmetric space associated with a noncompact semisimple Lie group. For any Hermitian matrices X and Y, So's matrix exponential formula asserts that there are unitary matrices U and V such thateX/2eYeX/2=eUXU⁎+VYV⁎. In other words, the Hermitian matrix log⁡(eX/2eYeX/2) lies in the sum of the unitary orbits of X and Y. So's result is also extended to a formula for adjoint orbits associated with a noncompact semisimple Lie group.

An uncertainty principle for spectral projections on rank one symmetric spaces of noncompact type

Let G be a noncompact semisimple Lie group with finite centre. Let \(X=G/K\) be the associated Riemannian symmetric space and assume that X is of rank one. The generalized spectral projections associated to the Laplace-Beltrami operator are given by \(P_{\lambda }f =f*\Phi _{\lambda }\), where \(\Phi _{\lambda }\) are the elementary spherical functions on X. In this paper, we prove an Ingham type uncertainty principle for \(P_{\lambda }f\). Moreover, similar results are obtained in the case of generalized spectral projections associated to Dunkl Laplacian.

Properly proximal groups and their von Neumann algebras

We introduce a wide class of countable groups, called properly proximal, which contains all non-amenable bi-exact groups, all non-elementary convergence groups, and all lattices in non-compact semi-simple Lie groups, but excludes all inner amenable groups. We show that crossed product II$_1$ factors arising from free ergodic probability measure preserving actions of groups in this class have at most one weakly compact Cartan subalgebra, up to unitary conjugacy. As an application, we obtain the first $W^*$-strong rigidity results for compact actions of $SL_d(\mathbb Z)$ for $d \geq 3$.

Characterization of eigenfunctions of the Laplace–Beltrami operator using Fourier multipliers

Let X be a rank one Riemannian symmetric space of noncompact type and Δ be the Laplace–Beltrami operator of X. The space X can be identified with the quotient space G/K where G is a connected noncompact semisimple Lie group of real rank one with finite center and K is a maximal compact subgroup of G. Thus G acts naturally on X by left translations. Through this identification, a function or measure on X is radial (i.e. depends only on the distance from eK), when it is invariant under the left-action of K. We consider right-convolution operators Θ on functions f on X defined by, Θ:f↦f⁎μ where μ is a radial (possibly complex) measure on X. These operators will be called multipliers. In particular Θ is a radial average when μ is a radial probability measure. Notable examples of radial averages are ball, sphere and annular averages and the heat operator. In this paper we address problems of the following type: Fix a multiplier, in particular an averaging operator Θ. Suppose that {fk}k∈Z is a bi-infinite sequence of functions on X such that for all k∈Z, Θfk=Afk+1 and ‖fk‖<M for some constants A∈C, M>0 and a suitable norm ‖⋅‖. From this hypothesis, we try to infer that f0, hence every fk, is an eigenfunction of Δ.

Parabolic Verma Modules and Invariant Differential Operators

In the present paper we continue the project of systematic classification and construction of invariant differential operators for non-compact semisimple Lie groups. This time we make the stress on one of the main building blocks, namely the Verma modules and the corresponding parabolic subalgebras. In particular, we start the study of the relation between the parabolic subalgebras of real semisimple Lie algebras and of their complexification. Two cases are given in more detail: the conformal algebra of 4D Minkowski space-time and the minimal parabolics of classical real semisimple Lie algebras.

On the volume of orbifold quotients of symmetric spaces

Key to H. C. Wang's quantitative study of Zassenhaus neighbourhoods of non-compact semisimple Lie groups are two constants that depend on the root system of the corresponding Lie algebra. This article extends the list of values for Wang's constants to the exceptional Lie groups and also removes their dependence on dimension. The first application is an improved upper sectional curvature bound for a canonical left-invariant metric on a semisimple Lie group. The second application is an explicit uniform positive lower bound for arbitrary orbifold quotients of a given irreducible symmetric space of non-compact type.

Homogeneous almost-Kähler manifolds and the Chern–Einstein equation

Given a non-compact semisimple Lie group G we describe all homogeneous spaces G / L carrying an invariant almost-Kahler structure $$(\omega ,J)$$ . When L is abelian and G is of classical type, we classify all such spaces which are Chern–Einstein, i.e. which satisfy $$\rho = \lambda \omega $$ for some $$\lambda \in {\mathbb {R}}$$ , where $$\rho $$ is the Ricci form associated to the Chern connection.

Twisted conjugacy and quasi-isometric rigidity of irreducible lattices in semisimple lie groups

Let G be a non-compact semisimple Lie group with finite centre and finitely many connected components. We show that any finitely generated group Γ which is quasi-isometric to an irreducible lattice in G has the R∞-property, namely, that there are infinitely many ϕ-twisted conjugacy classes for every automorphism ϕ of Γ. Also, we show that any lattice in G has the R∞-property, extending our earlier result for irreducible lattices.

Cellular homology of real flag manifolds

Let FΘ=G∕PΘ be a generalized flag manifold, where G is a real non-compact semi-simple Lie group and PΘ a parabolic subgroup. A classical result says the Schubert cells, which are the closure of the Bruhat cells, endow FΘ with a cellular CW structure. In this paper we exhibit explicit parametrizations of the Schubert cells by closed balls (cubes) in Rn and use them to compute the boundary operator ∂ for the cellular homology. We recover the result obtained by Kocherlakota [1995], in the setting of Morse Homology, that the coefficients of ∂ are 0 or ±2 (so that Z2-homology is freely generated by the cells). In particular, the formula given here is more refined in the sense that the ambiguity of signals in the Morse–Witten complex is solved.

Characteristic functions of semigroups in semi-simple Lie groups

Abstract Let G be a noncompact semi-simple Lie group with Iwasawa decomposition G = K ⁢ A ⁢ N {G=KAN} . For a semigroup S ⊂ G {S\subset G} with nonempty interior we find a domain of convergence of the Helgason–Laplace transform I S ⁢ ( λ , u ) = ∫ S e λ ⁢ ( 𝖺 ⁢ ( g , u ) ) ⁢ 𝑑 g {I_{S}(\lambda,u)=\int_{S}e^{\lambda(\mathsf{a}(g,u))}\,dg} , where dg is the Haar measure of G, u ∈ K {u\in K} , λ ∈ 𝔞 ∗ {\lambda\in\mathfrak{a}^{\ast}} , 𝔞 {\mathfrak{a}} is the Lie algebra of A and g ⁢ u = k ⁢ e 𝖺 ⁢ ( g , u ) ⁢ n ∈ K ⁢ A ⁢ N {gu=ke^{\mathsf{a}(g,u)}n\in KAN} . The domain is given in terms of a flag manifold of G written 𝔽 Θ ⁢ ( S ) {\mathbb{F}_{\Theta(S)}} called the flag type of S, where Θ ⁢ ( S ) {\Theta(S)} is a subset of the simple system of roots. It is proved that I S ⁢ ( λ , u ) &lt; ∞ {I_{S}(\lambda,u)&lt;\infty} if λ belongs to a convex cone defined from Θ ⁢ ( S ) {\Theta(S)} and u ∈ π - 1 ⁢ ( 𝒟 Θ ⁢ ( S ) ⁢ ( S ) ) {u\in\pi^{-1}(\mathcal{D}_{\Theta(S)}(S))} , where 𝒟 Θ ⁢ ( S ) ⁢ ( S ) ⊂ 𝔽 Θ ⁢ ( S ) {\mathcal{D}_{\Theta(S)}(S)\subset\mathbb{F}_{\Theta(S)}} is a B-convex set and π : K → 𝔽 Θ ⁢ ( S ) {\pi:K\rightarrow\mathbb{F}_{\Theta(S)}} is the natural projection. We prove differentiability of I S ⁢ ( λ , u ) {I_{S}(\lambda,u)} and apply the results to construct of a Riemannian metric in 𝒟 Θ ⁢ ( S ) ⁢ ( S ) {\mathcal{D}_{\Theta(S)}(S)} invariant by the group S ∩ S - 1 {S\cap S^{-1}} of units of S.

On density of infinite subsets II: Dynamics on homogeneous spaces

Let $G$ be a noncompact semisimple Lie group, $\Gamma$ be an irreducible cocompact lattice in $G$, and $P<G$ be a minimal parabolic subgroup. We consider the dynamics of $P$ acting on $G/\Gamma$ by left translation. For any infinite subset $A\subset G/\Gamma$, we show that, for any $\epsilon>0$, there is a $g\in P$ such that $gA$ is $\epsilon$-dense. We also prove a similar result for certain discrete group actions on $\mathbb T^n$.

Dynamics on flag manifolds: domains of proper discontinuity and cocompactness

For noncompact semisimple Lie groups $G$ we study the dynamics of the actions of their discrete subgroups $\Gamma<G$ on the associated partial flag manifolds $G/P$. Our study is based on the observation that they exhibit also in higher rank a certain form of convergence type dynamics. We identify geometrically domains of proper discontinuity in all partial flag manifolds. Under certain dynamical assumptions equivalent to the Anosov subgroup condition, we establish the cocompactness of the $\Gamma$-action on various domains of proper discontinuity, in particular on domains in the full flag manifold $G/B$. We show in the regular case (of $B$-Anosov subgroups) that the latter domains are always nonempty if if $G$ has (locally) at least one noncompact simple factor not of the type $A_1, B_2$ or $G_2$.

Weight distribution of Iwasawa projection

In the setting of a connected noncompact semisimple Lie group with a finite center, we prove a property of the Iwasawa decomposition, and use it to show that, in Kostant's convexity result [7], the Iwasawa projection puts almost all its weight near a single vertex of the convex hull. As another application, we will revisit the results of Huang–Tam [10,5] on the iteration of group elements under the Cartan and the Iwasawa decompositions.

A family of Markov processes in maximal compact subgroups of a semisimple Lie groups

We propose and define a family of marked point processes in noncompact semisimple Lie groups. We first generate Lévy processes via marked point processes by using jump–diffusion processes. Then we build a family of Markov processes in a maximal compact subgroup of a given semisimple Lie group.

Proper affine actions on semisimple Lie algebras

For any noncompact semisimple real Lie group $G$, we construct a group of affine transformations of its Lie algebra $\mathfrak{g}$ whose linear part is Zariski-dense in $\operatorname{Ad} G$ and which is free, nonabelian and acts properly discontinuously on $\mathfrak{g}$.

Degrees of maps between locally symmetric spaces

Let X be a locally symmetric space Γ\\G/K where G is a connected non-compact semisimple real Lie group with trivial centre, K is a maximal compact subgroup of G, and Γ⊂G is a torsion-free irreducible lattice in G. Let Y=Λ\\H/L be another such space having the same dimension as X. Suppose that real rank of G is at least 2. We show that any f:X→Y is either null-homotopic or it is homotopic to a covering projection of degree an integer that depends only on Γ and Λ. As a corollary we obtain that the set [X,Y] of homotopy classes of maps from X to Y is finite.We obtain results on the (non-)existence of orientation reversing diffeomorphisms on X as well as the fixed point property for X.

Loops in Noncompact Groups of Hermitian Symmetric Type and Factorization

In previous work with Pittmann-Polletta, we showed that a loop in a simply connected compact Lie group has a unique Birkhoff (or triangular) factorization if and only if the loop has a unique root subgroup factorization (relative to a choice of a reduced sequence of simple reflections in the affine Weyl group). In this paper our main purpose is to investigate Birkhoff and root subgroup factorization for loops in a noncompact type semisimple Lie group of Hermitian symmetric type. In previous work we showed that for a constant loop there is a unique Birkhoff factorization if and only if there is a root subgroup factorization. However for loops, while a root subgroup factorization implies a unique Birkhoff factorization, the converse is false. As in the compact case, root subgroup factorization is intimately related to factorization of Toeplitz determinants.

Classification of Invariant Differential Operators for Non-Compact Lie Algebras via Parabolic Relations

In the present paper we review the progress of the project of classification and construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we called earlier 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduced recently the new notion of parabolic relation between two non-compact semisimple Lie algebras G and G' that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra E7(7) which is parabolically related to the CLA E7(−25). Other interesting examples are the orthogonal algebras so(p, q) all of which are parabolically related to the conformal algebra so(n, 2) with p + q = n + 2, the parabolic subalgebras including the Lorentz subalgebra so(n − 1,1) and its analogs so(p − 1, q − 1). Further we consider the algebras sl(2n, ℝ) and for n = 2k the algebras su* (4k) which are parabolically related to the CLA su(n,n). Further we consider the algebras sp(r,r) which are parabolically related to the CLA sp(2r, ℝ). We consider also E6(6) and E6(2) which are parabolically related to the hermitian symmetric case E6(-14),

Growth estimates for orbits of self-adjoint groups

Let $$G$$ denote a closed, connected, self-adjoint, noncompact subgroup of $$GL(n,\mathbb R )$$ , and let $$d_{R}$$ and $$d_{L}$$ denote respectively the right and left invariant Riemannian metrics defined by the canonical inner product on $$M(n,\mathbb R ) = T_{I} GL(n,\mathbb R )$$ . Let $$v$$ be a nonzero vector of $$\mathbb R ^{n}$$ such that the orbit $$G(v)$$ is unbounded in $$\mathbb R ^{n}$$ . Then the function $$g \rightarrow d_{R}(g, G_{v})$$ is unbounded, where $$G_{v} = \{g \in G : g(v) = v \}$$ , and we obtain algebraically defined upper and lower bounds $$\lambda ^{+}(v)$$ and $$\lambda ^{-}(v)$$ for the asymptotic behavior of the function $$\frac{log|g(v)|}{d_{R}(g, G_{v})}$$ as $$d_{R}(g, G_{v}) \rightarrow \infty $$ . The upper bound $$\lambda ^{+}(v)$$ is at most 1. The orbit $$G(v)$$ is closed in $$\mathbb R ^{n} \Leftrightarrow \lambda ^{-}(w)$$ is positive for some w $$\in G(v)$$ . If $$G_{v}$$ is compact, then $$g \rightarrow |d_{R}(g,I) - d_{L}(g,I)|$$ is uniformly bounded in $$G$$ , and the exponents $$\lambda ^{+}(v)$$ and $$\lambda ^{-}(v)$$ are sharp upper and lower asymptotic bounds for the functions $$\frac{log|g(v)|}{d_{R}(g,I)}$$ and $$\frac{log|g(v)|}{d_{L}(g,I)}$$ as $$d_{R}(g,I) \rightarrow \infty $$ or as $$d_{L}(g,I) \rightarrow \infty $$ . However, we show by example that if $$G_{v}$$ is noncompact, then there need not exist asymptotic upper and lower bounds for the function $$\frac{log|g(v)|}{d_{L}(g, G_{v})}$$ as $$d_{L}(g, G_{v}) \rightarrow \infty $$ . The results apply to representations of noncompact semisimple Lie groups $$G$$ on finite dimensional real vector spaces. We compute $$\lambda ^{+}$$ and $$\lambda ^{-}$$ for the irreducible, real representations of $$SL(2,\mathbb R )$$ , and we show that if the dimension of the $$SL(2,\mathbb R )$$ -module $$V$$ is odd, then $$\lambda ^{+} = \lambda ^{-}$$ on a nonempty open subset of $$V$$ . We show that the function $$\lambda ^{-}$$ is $$K$$ -invariant, where $$K = O(n,\mathbb R ) \cap G$$ . We do not know if $$\lambda ^{-}$$ is $$G$$ -invariant.

A contraction of the principal series by Berezin-Weyl quantization

We study a contraction of the principal series representations of a noncompact semisimple Lie group to the unitary irreducible representations of its Cartan motion group by means of the Berezin-Weyl quantization on the coadjoint orbits associated with these representations.