The Union of Compact Subgroups of an Analytic Group

The main concern is the existence of a maximal compact normal subgroup K in a locally compact group G, and whether or not G/K is a Lie group.G has a maximal compact subgroup if and only if G/Go has.Maximal compact subgroups of totally disconnected groups are open.If the bounded part of G is compactly generated, then G has a maximal compact normal subgroup K and if B(G) is open, then G/K is Lie.Generalized FCgroups, compactly generated type I IN-groups, and Moore groups share the same properties.

The main concern is the existence of a maximal compact normal subgroup K in a locally compact group G, and whether or not G/K is a Lie group.G has a maximal compact subgroup if and only if G/Go has.Maximal compact subgroups of totally disconnected groups are open.If the bounded part of G is compactly generated, then G has a maximal compact normal subgroup K and if B(G) is open, then G/K is Lie.Generalized FCgroups, compactly generated type I IN-groups, and Moore groups share the same properties.

Our main interest is the existence of maximal compact normal subgroups of locally compact topological groups and its relation to compactly generated subgroups. If a topological group $G$ has a compact normal subgroup $K$ such that $G/K$ is a Lie group and every closed subgroup of $G$ is compactly generated, we call $G$ an $\mathcal {H}(c)$-group. If $G$ has a maximal compact normal subgroup $K$ such that $G/K$ is a Lie group, we call $G$ an $\mathcal {H}$-group. If $G$ is an $\mathcal {H}(c)$-group, then $G$ is a hereditary $\mathcal {H}$-group in the sense that every closed subgroup is an $\mathcal {H}$-group. If $H$ is a closed normal subgroup of $G$ and both $H,G/H$ are $\mathcal {H}(c)$-groups, then $G$ is an $\mathcal {H}(c)$-group. A corollary of this is that a compactly generated solvable group whose characteristic open subgroups are compactly generated is an $\mathcal {H}$-group. If $G$ has a compactly generated closed normal subgroup $F$ such that both $F/{F_0}G/F$ are $\mathcal {H}$-groups, then $G$ is an $\mathcal {H}$-group.

Our main interest is the existence of maximal compact normal subgroups of locally compact topological groups and its relation to compactly generated subgroups. If a topological group G G has a compact normal subgroup K K such that G / K G/K is a Lie group and every closed subgroup of G G is compactly generated, we call G G an H ( c ) \mathcal {H}(c) -group. If G G has a maximal compact normal subgroup K K such that G / K G/K is a Lie group, we call G G an H \mathcal {H} -group. If G G is an H ( c ) \mathcal {H}(c) -group, then G G is a hereditary H \mathcal {H} -group in the sense that every closed subgroup is an H \mathcal {H} -group. If H H is a closed normal subgroup of G G and both H , G / H H,G/H are H ( c ) \mathcal {H}(c) -groups, then G G is an H ( c ) \mathcal {H}(c) -group. A corollary of this is that a compactly generated solvable group whose characteristic open subgroups are compactly generated is an H \mathcal {H} -group. If G G has a compactly generated closed normal subgroup F F such that both F / F 0 G / F F/{F_0}G/F are H \mathcal {H} -groups, then G G is an H \mathcal {H} -group.

Introduction. It is well known that differentiable actions of compact Lie groups are far more regular than those of the non-compact Lie groups and the behavior of compact Lie group actions are comparatively much more well understood than those of the non-compact Lie groups. Of course, one of the most basic and important cases of non-compact Lie group actions are those Rl-actions which are naturally associated with ordinary differential equations [6 ]. So far, much important research has been done in that direction. However, knowledge about R'-actions still seems rather inadequate. Since the group R1 is a simply connected one-dimensional Lie group and it contains no non-trivial compact subgroups, neither the theory of Lie groups nor knowledge about compact Lie group actions are of any help in the study of R'-actions. In this note, we try to look at the other extreme case of actions of large semi-simple non-compact Lie groups. The situation here is quite different from the study of RI-actions. Roughly speaking, the problem is more complicated on the one hand, and has much more built in structures on the other hand. For example, one naturally expects that the profound theory of semisimple Lie groups and the knowledge of compact Lie grfoup actions might be useful at least for some special but natural cases. Due to the fact that one knows so little about the behavior of differentiable actions of non-compact Lie groups in general, it seems to be worthwhile to investigate some special but natural cases such as semi-simple Lie group actions on euclidean spaces. Let G be a semi-simple non-compact Lie group. It is well known that there exists a maximal compact subgroup H which is unique up to conjugacy. For a given differentiable manifold M'4, let r(GMm]Ii4) and r (H, Mm) be the set of equivalence classes of differentiable actions of G and H on Ml', respectively. To restrict those G-actions on MIlm to H induces a natural luap i !: r (G, Mm) r (H, Mm) which is independent of the choice of the maximal compact subgroup H. A reasonable way to study G-actions on Mm is to investigate the image and the kernel of the above map. To be more precise,

Our main concern is the existence of maximal compact subgroups in a locally compact topological group. If G is a locally compact group such that P(G/G o), the set of periodic points of G/Go, is a compact subgroup of G/Go, than G has maximal compact subgroups K such that G/N is a Lie group where N = ∩K, the intersection of the collection K of all maximal compact subgroups of G. Also every compact subgroup of G is contained in a maximal compact subgroup. We given an example of a discrete group which has maximal finite subgroup and has finite subgroups not contained in maximal finite subgroups. We note that the above result on P(G/Go) is an extension of the well-known corresponding result for almost connected groups.

We are concerned with conditions under which a locally compact group $G$ has a maximal compact normal subgroup $K$ and whether or not $G/K$ is a Lie group. If $G$ has small compact normal subgroups $K$ such that $G/K$ is a Lie group, then $G$ is pro-Lie. If in $G$ there is a collection of closed normal subgroups $\{ {H_\alpha }\}$ such that $\cap {H_\alpha } = e$ and $G/{H_\alpha }$ is a Lie group for each $\alpha$, then $G$ is a residual Lie group. We determine conditions under which a residual Lie group is pro-Lie and give an example of a residual Lie group which is not embeddable in a pro-Lie group.

Let G be a connected, semisimple Lie group. In Harish-Chandra's work on the Plancherel formula, the discrete series of irreducible unitary representations [5] plays a crucial role. Roughly speaking, besides the discrete series itself, the representations which enter the Plancherel decomposition of L2(G) can all be built up from discrete series representations of subgroups of G. Among the various irreducible unitary representations, the discrete series representations are therefore of particular importance. The main technique in the representation theory of compact Lie groups is to investigate the restriction of any given representation to a maximal torus. Similarl~r to understand the structure of an irreducible representation of a noncompact semisimple Lie group, one may try to determine its restriction to a maximal compact subgroup. Blattner's conjecture predicts how the discrete series representations should break up under the action of a maximal compact subgroup. It formally resembles the formula for the multiplicity of a weight for finite-dimensional representations. A companion statement to Blattner's conjecture, analogous to the theorem of the highest weight, characterizes discrete series representations, up to infinitesimal equivalence, in terms of their restriction to a maximal compact subgroup [13]. Once and for all, we make the assumption that G contains a compact Cartan subgroup. Exactly in this situation G has a non-empty discrete series [5]. Let K be a maximal compact subgroup, with HcK. The Lie algebras of G, K, H will be written as go, ~0, bo, and their complexifications as g, [, b. A non-zero root of (g, b) is called compact or noncompact, depending on whether or not its root space lies in L We denote the sets of compact and noncompact roots by, respectively, q~ and 4"; ~ = ~c w 4" is thus the setof all non-zero roots of (g, b). The Killing form of g induces an inner product ( , ) on ib*, the space of linear functions on b which assume imaginary values on b0An element ~teib* is said to be singular if it is perpendicular to at least one ct~q~, and nonsingular otherwise. The differentials of the characters of H form a lattice A cib*, which contains ~b. Since

We are concerned with conditions under which a locally compact group G G has a maximal compact normal subgroup K K and whether or not G / K G/K is a Lie group. If G G has small compact normal subgroups K K such that G / K G/K is a Lie group, then G G is pro-Lie. If in G G there is a collection of closed normal subgroups { H α } \{ {H_\alpha }\} such that ∩ H α = e \cap {H_\alpha } = e and G / H α G/{H_\alpha } is a Lie group for each α \alpha , then G G is a residual Lie group. We determine conditions under which a residual Lie group is pro-Lie and give an example of a residual Lie group which is not embeddable in a pro-Lie group.

A topological group G is pro-Lie if G has small compact normal subgroups K such that G/K is a Lie group. A locally compact group G is an L-group if, for every neighborhood U of the identity and compact set C, there is a neighborhood V of the identity such that gHg1 n C C U for every g C G and every subgroup H C V. We obtain characterizations of pro-Lie groups and make several applications. For example, every compactly generated L-group is pro-Lie and a compactly generated group which can be embedded (by a continuous isomorphism) in a pro-Lie group is pro-Lie. We obtain related results for factor groups, nilpotent groups, maximal compact normal subgroups, and generalize a theorem of Hofmann, Liukkonen, and Mislove [4]. Introduction. We divide the paper into two sections. In ?1 we consider some rather general properties of locally compact groups and establish their relationships to pro-Lie groups. In ?2 we obtain some related results and consider pro-Lie groups in more special settings. We obtain several characterizations of pro-Lie groups, the last one in terms of the bounded part and periodic part of the group. In this paper we are concerned only with those groups which are locally compact. The following background is taken directly from the referee's report. The application of Lie group theory is the basis of most of the finer results on the structure theory of locally compact groups. Historically, the first half of this century saw topological group theory concentrate on the interaction of Lie group theory and the general theory of compact and locally compact groups. The guiding principle was one of Hilbert's problems formulated at the turn of the century (the famed number FIVE). Even before this program was completed, Iwasawa published a key paper in 1949 on the structure theory of locally compact groups which are projective limits of Lie groups, i.e. those groups which now are called somewhat unfortunately from the view point of good English-pro-Lie groups. The vitality of Iwasawa's ideas continues to influence writers in the area up to these days, as is, e.g., exemplified by this paper. The fine-structure theory of locally compact groups flourished through the sixties. While this work did not show the raw power of penetration of the work of Gleason, Montgomery, and Yamabe, which in the early fifties led to the solution of Hilbert's Fifth Problem, it contributed much to the insight into the structure of more special classes of locally compact groups through the contributions of Grosser, Hofmann, Moskowitz, Mostert and their followers and students. In the meantime one observes a steady, although somewhat slower, flow of results on locally compact groups, and their structure theory springs from the work of researchers in the field, while more Received by the editors November 23, 1983 and, in revised form, April 9, 1984. 1980 Mathematics S*ect Clwsification. Primary 22D05. (?)1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page

A topological group $G$ is pro-Lie if $G$ has small compact normal subgroups $K$ such that $G/K$ is a Lie group. A locally compact group $G$ is an $L$-group if, for every neighborhood $U$ of the identity and compact set $C$, there is a neighborhood $V$ of the identity such that $gH{g^{ - 1}} \cap C \subset U$ for every $g \in G$ and every subgroup $H \subset V$. We obtain characterizations of pro-Lie groups and make several applications. For example, every compactly generated $L$-group is pro-Lie and a compactly generated group which can be embedded (by a continuous isomorphism) in a pro-Lie group is pro-Lie. We obtain related results for factor groups, nilpotent groups, maximal compact normal subgroups, and generalize a theorem of Hofmann, Liukkonen, and Mislove [4].

This chapter is chiefly devoted to a fundamental theorem which asserts, roughly, that if G is a Lie group with finitely many connected components, then G has maximal compact subgroups, is homeomorphic to the product of any one of them by a euclidean space, and any two maximal compact subgroups of G are conjugate by an inner automorphism. The precise statement, and two corollaries, are given in §1, the proof in §2. The latter makes use of a theorem of Iwasawa on extensions of vector groups by compact groups, and of several results obtained in the previous chapters, notably the decomposition Ad g = ωK × P, and the conjugacy of maximal compact subgroups in Aut <math display='block'> <mi mathvariant='fraktur'>g</mi> </math> $$\mathfrak{g}$$ , where <math display='block'> <mi mathvariant='fraktur'>g</mi> </math> $$\mathfrak{g}$$ is a semisimple Lie algebra. In order to make this chapter more self-contained, we give in §3 alternative proofs of those statements on semi-simple Lie groups, which do not make use of Riemannian geometry, and reproduce a proof of the Iwasawa theorem in §4.

The set of all linear transformations on \(\mathbb {R}^{2m}\), preserving its b-symplectic structure, is a Lie group. This group has two components and its identity component contains the unitary group \(U(m-1)\) as maximal compact subgroup. We obtain a unitary representation for the identity component. At last some results on embedding by this transformations are presented.

The action of a torus group $T$ on a symplectic toric manifold $(M,\omega)$ often extends to an effective action of a (non-abelian) compact Lie group $G$. We may think of $T$ and $G$ as compact Lie subgroups of the symplectomorphism group $Symp(M,\omega)$ of $(M,\omega)$. On the other hand, $(M,\omega)$ is determined by the associated moment polytope $P$ by the result of Delzant. Therefore, the group $G$ should be estimated in terms of $P$ or we may say that a maximal compact Lie subgroup of $Symp(M,\omega)$ containing the torus $T$ should be described in terms of $P$. In this paper, we introduce a root system $R(P)$ associated to $P$ and prove that any irreducible subsystem of $R(P)$ is of type A and the root system $\Delta(G)$ of the group $G$ is a subsystem of $R(P)$ (so that $R(P)$ gives an upper bound for the identity component of $G$ and any irreducible factor of $\Delta(G)$ is of type A). We also introduce a homomorphism from the normalizer of $T$ in $G$ to an automorphism group $Aut(P)$ of $P$, which detects the connected components of $G$. Finally we find a maximal compact Lie subgroup $G_{\max}$ of $Symp(M,\omega)$ containing the torus $T$.

Let $S$ be a complex reductive group acting holomorphically on a complex Lie group $N$ via holomorphic automorphisms. Let $K(S)\subset S$ be a maximal compact subgroup. The semidirect product $G := N\rtimes K(S)$ acts on $N$ via biholomorphisms. We give an explicit description of the isomorphism classes of $G$-equivariant almost holomorphic hermitian principal bundles on $N$. Under the assumption that there is a central subgroup $Z= \text{U}(1)$ of $K(S)$ that acts on $\text{Lie}(N)$ as multiplication through a single nontrivial character, we give an explicit description of the isomorphism classes of $G$-equivariant holomorphic hermitian principal bundles on $N$.