Ultra-irreducibility of induced representations of semidirect products

Let G be an analytic group.Let Q(G) be the union of all compact subgroups of G .We give a necessary and sufficient condition for 2(G) to be dense in G in terms of the action of a maximal compact torus T of G on the nilradical TV of G.Let F be a locally compact group.Let Q(F) be the union of all compact subgroups of F .We study the problem: when Q(F) is dense in F. If F is not connected, the problem is too broad to have any meaningful answers.On the other hand, if F is almost connected, i.e., F Fo is compact where Fo is the identity component of F , then the problem is quickly reduced to the case where F is a Lie group with finitely many components.This is so because an almost connected locally compact F has a maximal compact normal subgroup M so that F M is a Lie group with finitely many components.It is easy to see that 2(F) is dense in F if and only if Q(F/A/) is dense in F/M.Let G = F M. Let C70 be the identity component of G. Since the identity component Go of G is an open subgroup, so Q(C7) n Go is dense in Go when Ci(G) is dense in G (the converse is also true, cf.Theorem 2.10).Therefore, for most of this note we shall assume that G is an analytic group.Now, let G be an analytic group with Q(G) dense in G. Let M be the maximal compact normal subgroup of G. Again, Ci(G) is dense in G if and only if Q(G/M) is dense in G/M, so we may assume that M is trivial.Let A be the nilradical of G, i.e., the maximal analytic nilpotent normal subgroup of G. Then N is simply connected since M is trivial.Furthermore, by an argument due to Djokovic [1] we can show that A is uniform in G.This implies that G is a semidirect product A K with K a compact analytic group.Hence K acts on A as a group of automorphisms.The purpose of the present note is to show the following statement.Theorem 2.7.Let G be a semidirect product N K with A a simply connected analytic nilpotent group and K a compact analytic group.Let T be a maximal torus of K. Then Q(t7) is dense in G if and only if the only element in N fixed by T is the identity element.Another characterization of Q(C7) being dense in G is the following condition._

Maximal representations into Lie groups of Hermitian type have been introduced in [7], and further studied in [2,6,26]. All maximal representation are discrete embeddings, and spaces of maximal representations are unions of connected components of the character varieties, hence they provide examples of so-called higher Teichmüller spaces. Connected components of spaces of maximal representations have complicated topology which is not well understood.
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\nIn this thesis, we study classical Hermitian Lie groups of tube type and give a parametrization of spaces of decorated (maximal) representations of the fundamental group of a punctured surface into a Hermitian Lie group of tube type. Using this parametrization, we describe the topology and the structure of the spaces of maximal representations.
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\nIn the first chapter, we introduce coordinates on the space of Lagrangian decorated representations of the fundamental group of a surface with punctures into the symplectic group Sp(2n, R). These coordinates provide a noncommutative generalization of the parametrization of the space of representations into SL(2, R) given by V. Fock and A. Goncharov. The locus of positive coordinates maps to the space of decorated maximal representations. We use this to determine the homotopy type and the homeomorphism type of the space of decorated maximal representations, and when n = 2, to describe its finer structure as a smooth locus and kind of singularities.
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\nIn the second chapter, we study Hermitian Lie groups of tube type and their complexifications uniformly as Sp2(A) over some special real algebra A. We use this approach to describe the flag variety of such groups corresponding to a maximal parabolic subgroup, a maximal compact subgroup and different models of the symmetric space. For complexified groups this construction is new. Further, we introduce in these terms coordinates on the space of decorated maximal representations of the fundamental group of a punctured surface into a Hermitian Lie group of tube type and use them to determine the homotopy type and the homeomorphism type of the space of decorated maximal representations.

Given a Lie group $G$ and a lattice $\Gamma$ in $G$, a one-parameter subgroup $\phi$ of $G$ is said to be rigid if for any other one-parameter subgroup $\psi$, the flows induced by $\phi$ and $\psi$ on $\Gamma\backslash G$ (by right translations) are topologically orbit-equivalent only if they are affinely orbit-equivalent. It was previously known that if $G$ is a simply connected solvable Lie group such that all the eigenvalues of $\mathrm{Ad} (g) $, $g\in G$, are real, then all one-parameter subgroups of $G$ are rigid for any lattice in $G$. Here we consider a complementary case, in which the eigenvalues of $\mathrm{Ad} (g)$, $g\in G$, form the unit circle of complex numbers.Let $G$ be the semidirect product $N \rtimes M$, where $M$ and $N$ are finite-dimensional real vector spaces and where the action of $M$ on the normal subgroup $N$ is such that the center of $G$ is a lattice in $M$. We prove that there is a generic class of abelian lattices $\Gamma$ in $G$ such that any semisimple one-parameter subgroup $\phi$ (namely $\phi$ such that $\mathrm{Ad} (\phi_t)$ is diagonalizable over the complex numbers for all $t$) is rigid for $\Gamma$ (see Theorem 1.4). We also show that, on the other hand, there are fairly high-dimensional spaces of abelian lattices for which some semisimple $\phi$ are not rigid (see Corollary 4.3); further, there are non-rigid semisimple $\phi$ for which the induced flow is ergodic.

The first part of this chapter treats Lie algebras, beginning with definitions and many examples. The notions of solvable, nilpotent, radical, semisimple, and simple are introduced, and these notions are followed by a discussion of the effect of a change of the underlying field. The idea of a semidirect product begins the development of the main structural theorems for real Lie algebras—the iterated construction of all solvable Lie algebras from derivations and semidirect products, Lie's Theorem for solvable Lie algebras, Engel's Theorem in connection with nilpotent Lie algebras, and Cartan's criteria for solvability and semisimplicity in terms of the Killing form. From Cartan's Criterion for Semisimplicity, it follows that semisimple Lie algebras are direct sums of simple Lie algebras. Cartan's Criterion for Semisimplicity is used also to provide a long list of classical examples of semisimple Lie algebras. Some of these examples are defined in terms of quaternion matrices. Quaternion matrices of size n-by-n may be related to complex matrices of size 2n-by-2n. The treatment of Lie algebras concludes with a study of the finite-dimensional complex-linear representations of sl[(2, ℂ). There is a classification theorem for the irreducible representations of this kind, and the general representations are direct sums of irreducible ones. Section 10 contains a review of the elementary theory of Lie groups and their Lie algebras. The abstract theory as in Chevalley [1946] is summarized, and the correspondence is made with the concrete theory of closed linear groups, where the Lie algebra is obtained as the space of derivatives at t = 0 of smooth curves in the group passing through the identity at t = 0. The section ends with a discussion of the adjoint representation. The remainder of the chapter explores some aspects of the connection between Lie groups and Lie algebras. One aspect is the relationship between automorphisms and derivations. The derivations of a semisimple Lie algebra are inner, and consequently the identity component of the group of automorphisms of a semisimple Lie algebra consists of inner automorphisms. In addition, simply connected solvable Lie groups may be built one dimension at a time as semidirect products with ℝ1, and consequently they are diffeomorphic to Euclidean space. For simply connected nilpotent groups the exponential map is itself a diffeomorphism. The earlier long list of classical semisimple Lie algebras corresponds to a list of the classical semisimple Lie groups. The issue that needs attention for these groups is their connectedness, and this is proved by using the polar decomposition of matrices.

Let G be a connected Lie group which is a semidirect product of a compact subgroup K and a normal solvable subgroup S. Let Λ \Lambda be a character of S, and let M Λ {M_\Lambda } be the stabilizer of Λ \Lambda in K. Let [ H , Λ μ ] [H,{\Lambda _\mu }] be a finite-dimensional irreducible representation of the subgroup S M Λ S{M_\Lambda } on the complex vector space H. In this paper we consider the induced representations of G on various Banach spaces, and study their topological irreducibility. The basic method used consists in studying the irreducibility of the Lie algebra representations which arise on the linear subspaces of K-finite vectors. The latter question then can be reduced to the problem of determining when certain modules over certain commutative algebras are irreducible. The method discussed in this paper leads to two theorems giving sufficient conditions on the character Λ \Lambda that the induced representations be topologically irreducible. The question of infinitesimal equivalence of various induced representations is also discussed.

Cohomology of Semidirect Product Groups

There exist two types of semi-direct products between a Lie group G and a vector space V. The left semi-direct product, G⋉ V, can be constructed when G is equipped with a left action on V. Similarly, the right semi-direct product, G⋊ V, can be constructed when G is equipped with a right action on V. In this paper, we will construct a new type of semi-direct product, $$G \bowtie V$$ , which can be seen as the ‘sum’ of a right and left semi-direct product. We then parallel existing semi-direct product Euler-Poincaré theory. We find that the group multiplication, the Lie bracket, and the diamond operator can each be seen as a sum of the associated concepts in right and left semi-direct product theory. Finally, we conclude with a toy example and the group of 2-jets of diffeomorphisms above a fixed point. This final example has potential use in the creation of particle methods for problems on diffeomorphism groups.

In these notes we present some aspects of the basic theory on the geometry of a three-dimensional simply-connected Lie group X X endowed with a left invariant metric. This material is based upon and extends some of the results of Milnor in Curvatures of left invariant metrics on Lie groups . We then apply this theory to study the geometry of constant mean curvature H ≥ 0 H\geq 0 surfaces in X X , which we call H H -surfaces . The focus of these results on H H -surfaces concerns our joint on going research project with Pablo Mira and Antonio Ros to understand the existence, uniqueness, embeddedness and stability properties of H H -spheres in X X . To attack these questions we introduce several new concepts such as the H H -potential of X X , the critical mean curvature H ( X ) H(X) of X X and the notion of an algebraic open book decomposition of X X . We apply these concepts to classify the two-dimensional subgroups of X X in terms of invariants of its metric Lie algebra, as well as classify the stabilizer subgroup of the isometry group of X X at any of its points in terms of these invariants. We also calculate the Cheeger constant for X X to be Ch ( X ) = trace ( A ) (X)=\mbox {trace}(A) , when X = R 2 ⋊ A R X=\mathbb {R}^2\rtimes _A\mathbb {R} is a semidirect product for some 2 × 2 2\times 2 real matrix; this result is a special case of a more general theorem by Peyerimhoff and Samiou. We also prove that in this semidirect product case, Ch ( X ) = 2 H ( X ) = 2 I ( X ) (X)=2H(X)=2I(X) , where I ( X ) I(X) is the infimum of the mean curvatures of isoperimetric surfaces in X X . In the last section, we discuss a variety of unsolved problems for H H -surfaces in X X .

Orbit spaces of low-dimensional representations of classical and exceptional Lie groups are constructed and tabulated. We observe that the orbit spaces of some single irreducible representations (adjoints, second-rank symmetric and antisymmetric tensors of classical Lie groups, and the defining representations of F4 and E6) are warped polyhedrons with (locally) more protrudent boundaries corresponding to higher level little groups. The orbit spaces of two irreducible representations have different shapes. We observe that dimension and concavity of different strata are not sharply distinguished. We explain that the observed orbit space structure implies that a physical system tends to retain as much symmetry as possible in a symmetry breaking process. In Appendix A, we interpret our method of minimization in the orbit space in terms of conventional language and show how to find all the extrema (in the representation space) of a general group-invariant scalar potential monotonic in the orbit space. We also present the criterion to tell whether an extremum is a local minimum or maximum or an inflection point. In Appendix B, we show that the minimization problem can always be reduced to a two-dimensional one in the case of the most general Higgs potential for a single irreducible representation and to a three-dimensional one in the case of an even degree Higgs potential for two irreducible representations. We explain that the absolute minimum condition prompts the boundary conditions enough to determine the representation vector.

A limiting formula is given for the representation theory of the Cartan motion group associated to a Riemannian symmetric pair (G, K) in terms of the representation theory of G.Introduction.Let G be a connected Lie group with Lie algebra g, and H a closed subgroup with subalgebra b.The coset space G/H is called reductive [9, p. 389] if h admits an AdG(H) invariant complement m in g; i.e. a subspace m c g such thatIn this case we can form the semidirect product m "A H with respect to the adjoint action of H on m.In this paper we shall restrict ourselves to the case where G is semisimple with finite centre and (G, H) is a Riemannian symmetric pair [10, p. 209].Hence H is contained in the fixed point set Ha of an analytic involution a of G, it contains the identity component (Ha)e and Adc(H) is compact.Following custom, we write K and f rather than H and b in this instance.It is well known that K is compact [10, p. 252] and connected if G is noncompact.Furthermore f is the +1 eigenspace of doe.We make the natural choice for m, namely the -1 eigenspace V of dae.When G is noncompact dae is a Cartan involution.Then V is usually denoted p and g = f + /? is called a Cartan decomposition [10, p. 182].When G is compact one can choose a real form g0 of the complexification gc of g, and a Cartan decomposition g0 = k + /? such that V = ip, i.e. g = f + ip [10, V, 2].The semidirect product F X AT, in the situation described above, is called the Cartan motion group associated to the pair (G, K).The idea of relating the representation theories of V X K and G has been prevalent in the literature (cf.notably [11,13,18]).In particular, V XI K is a contraction of the Lie group G in the sense of [11], and there has been some interest in understanding the relationship between the representation theories of V X K and G that this implies (cf.particularly the footnote on p. 343 of [17]).The aim of this paper is to give such a precise relationship.A key ingredient involved is the global counterpart of the contraction of the Lie algebra of G to that of V X K.It is the family of smooth maps wx: VXK^G (vk) - expc(X;)A:.These are homomorphisms to within 0( X ) as X <-* 0.

Semidirect products and the Pukanszky condition

This article develops an equivariant symmetry for second-order kinematic systems on matrix Lie groups and uses this symmetry for observer design. The state of a second-order kinematic system on a matrix Lie group is naturally posed on the tangent bundle of the group with the inputs lying in the tangent of the tangent bundle known as the double-tangent bundle. We provide a simple parameterization of both the tangent bundle state-space and the input space (the fiber space of the double-tangent bundle) and then introduce a semidirect product group and group actions onto both the state and input spaces. We show that with the proposed group actions, the second-order kinematics are equivariant. An equivariant lift of the kinematics onto the symmetry group is derived and used to design nonlinear observers on the lifted state-space using nonlinear constructive design techniques. The observer design is specialized to kinematics on groups that themselves admit a semidirect product structure and include applications in rigid-body motion amongst others. A simulation based on an ideal hovercraft model verifies the performance of the proposed observer architecture.

Coherent state representations. A survey

A normal Hall subgroup $N$ of a group $G$ is a normal subgroup with its order coprime with its index. Schur-Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall [STACS 2009], we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem. A solution to the latter can be obtained by a mild extension of the singly exponential (in the number of coordinates) time algorithm for code isomorphism problem developed recently by Babai. Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups: given two representations rho and tau of a group $H$ over Z_p^d , p a prime, determine if there exists an automorphism phi:H -> H, such that the induced representation rho_phi=rho o phi and tau are equivalent, in time poly(|H|,p^d).

We study integrable systems on the semidirect product of a Lie group and its Lie algebra as the representation space of the adjoint action. Regarding the tangent bundle of a Lie group as phase space endowed with this semidirect product Lie group structure, we construct a class of symplectic submanifolds equipped with a Dirac bracket on which integrable systems (in the Adler–Kostant–Symes sense) are naturally built through collective dynamics. In doing so, we address other issues such as factorization, Poisson–Lie structures and dressing actions. We show that the procedure becomes recursive for some particular Hamilton functions, giving rise to a tower of nested integrable systems.