Differentiable Actions of Compact Connected Lie Groups: III

The theory of compact Lie transformation groups is a large and well researched area of mathematics, in which topological methods are widely used and which requires a separate survey. In this chapter we only touch upon certain problems of this theory. In §1 we shall survey the classical results of Gleason, Montgomery, Mostow, Palais, Samelson and other authors concerning the structure of of orbits of actions of compact Lie groups. We emphasize the differentiable aspects of the theory and a number of results can be formulated in a more general setting — for proper actions of not necessarily compact Lie groups. §2 is devoted to properties of differentiable invariants and almost-invariants of compact Lie groups. In §3 we consider complexifications of homogeneous spaces of compact Lie groups and factorizations of reductive algebraic groups. Surveys of many questions not considered here can be found in (Bourbaki 1982), (Bredon 1972), (Hsiang 1975), (Janich 1968), (Palais 1960), (Schultz 1984).

This chapter explores integration on a compact connected Lie group. One of the great advantages of working with a compact Lie group is the possibility of extending the notion of averaging from a finite group to the compact Lie group. If the compact Lie group is connected, then there exists a unique bi-invariant top-degree form with total integral 1, which simplifies the presentation of averaging. The averaging operator is useful for constructing invariant objects. For example, suppose a compact connected Lie group G acts smoothly on the left on a manifold M. Given any C∞ differential k-form ω‎ on M, by averaging all the left translates of ω‎ over G, one can produce a C∞ invariant k-form on M. As another example, on a G-manifold one can average all translates of a Riemannian metric to produce an invariant Riemann metric.

Given a pair $(M,X)$, where $X$ is a smooth submanifold in a closed smooth manifold $M$, we study the operation, which takes each operator $D$ on the ambient manifold to a certain operator on the submanifold. The latter operator is called the trace of $D$. More precisely, we study traces of operators, associated with actions of compact Lie groups on $M$. We show that traces of such operators are localized at special submanifolds in $X$ and study the structure of the traces on these submanifolds.

We expose explicit calculations of the spectra of the Laplace operators for smooth real or complex functions on all connected compact simple Lie groups of rank 3 with bi-invariant Riemannian metric and establish the relationship of the obtained formulas with number theory and integer-valued ternary and binary quadratic forms.

Introduction. 1. Hilbert's fifth problem and proper actions of Lie groups S. Illman. 2. Equivariant algebraic vector bundles over representations - a survey M. Masuda. 3. G-manifolds and G-vector bundles in algebraic, semialgebraic, and definable categories T. Kawakami. 4. Geometry of Finite Topological and equivariant finite topological spaces S. Kono, F. Ushitaki. 5. On the theory of homotopy representations. A survey I. Nagasaki. 6. Manifolds as fixed point sets of smooth compact Lie group actions K. Pawalowski. 7. Surgery and homotopy theory in study of the transformation groups Y. Kitada. 8. Kervaire's obstructions of free actions of finite cyclic groups on homotopy spheres Y. Kitada. 9. The Burnside ring revisited M. Morimoto. 10. Multiplicative stabilization and transformation groups S. Kwasik, R. Schultz. 11. Symmetries on manifolds, deformations and rational homotopy: a survey M. Raussen. 12. Rigidity of codimension one locally free actions of solvable Lie groups N. Tsuchiya, A. Yamakawa. 13. Smooth actions of non-compact semi-simple Lie groups F. Uchida, K. Mukoyama. 14. Hamiltonian group actions and equivariant indices T. Takakura. 15. Controlled methods in equivariant topology, a survey E. Pedersen. Index.

Let (M be a group with unit element e. An element g C (M is a k-th root of unity, for some integer ki ? 2, if gk e. If, in addition, gi j e for all j 1,2, 2 , Ic-i, then g is a primitive k-th root. The purpose of this paper is to show that, when (M is a compact connected Lie group, the structure of primitive roots of unity is, to a considerable extent, determined by the topology of (M and hence is independent of the algebraic structure on (M. Our approach will be, instead of considering only Lie groups, to study a larger class of objects. By an H-manifold we shall mean a triple (M. m, e) where M is a compact connected topological manifold without boundary, e C M, and m: M X M -> M is a map such that mr(x, e) rm(e,x) ==x for all xC M. Define in1: M --VI by m, (x) ==x and, for 7A>2, define Mk(X) m (X, mk-l (X) ). A primitive 1-th root of unity in the H-manifold (M, m, e) is a point x C M such that mk(x) = e but mj (x) 7& e for all j < k. We will studyprimitive roots of unity in this setting. We remark that the subject is of interest, independent of its application to Lie groups. Recent results by Hilton, Belfi [1], Morgan [6], and Zabrodsky produce many new classes of H-manifolds which are not homeomorphic to any Lie group or even, in some cases, of the same homotopy type as any Lie group. We shall show that the existence of primitive roots of unity in an Hmanifold (M, m, e) is independent of the choice of the map m in the sense that:

Essential results about actions of compact Lie groups on connected manifolds are generalized to proper actions of arbitrary groups on connected cohomology manifolds. Slices are replaced by certain fiber bundle structures on orbit neighborhoods. The group dimension is shown to be effectively finite. The orbits of maximal dimension form a dense open connected subset. If some orbit has codimension at most $2$, then the group is effectively a Lie group.

In this article we study the behavior of harmonic maps from compact connected Lie groups with bi-invariant metrics into a Hilbert orthogonal group. In particular, we will demonstrate that any such harmonic map always has image contained within someO(n),n<∞. Since homomorphisms are a special subset of the harmonic maps we get as a corollary an extension of the Peter-Weyl theorem, namely, that every representation of a connected compact Lie group is finite dimensional.

Equivariant Alexander-Spanier cohomology for actions of compact Lie groups.

We classify isometric actions of compact Lie groups on quaternionic-Khler projective spaces with vanishing homogeneity rank.We also show that in general they are not quaternion-coisotropic.

Compact Lie group action with the continuous Rokhlin property

Let G be a compact connected Lie group and assume a choice of maximal torus and positive roots has been made. Given a dominant weight $\lambda$, the Borel-Weil Theorem shows how to construct a holomorphic line bundle on whose sections G acts so that the holomorphic sections provide a realization of the irreducible representation of G with highest weight $\lambda$. This paper studies the G-module structure of the space $\Gamma$ of square integrable sections of the Borel-Weil line bundle. It is found that $\Gamma = {\lim _{n \to \infty }}\Gamma (n)$, where $\Gamma (n) \subset \Gamma (n + 1) \subset \Gamma$ and $\Gamma (n)$ is isomorphic, as G-module, to \[ V(\lambda + n\lambda ) \otimes V(n{\lambda ^\ast }),\] where $V(\mu )$ denotes the irreducible representation of highest weight $\mu$, ’+’ is the Cartan semigroup operation, and ’$^\ast$’ is the contragredient operation. Similar formulas hold for powers of the Borel-Weil line bundle.

Borsuk-Ulam type theorems for arbitrary compact Lie group actions are proven. The transfer plays a major role in this approach.

We consider Sobolev problems with nonlocal boundary conditions associated with an action of a compact Lie group. We find a natural conditions of ellipticity of such problems, obtain the corresponding finiteness theorem, and give the index formula.

Here, a space is perfectly normal if it is normal, and every open subspace is a countable union of closed subspaces. Transfer maps for actions of finite groups were defined by Floyd [10], and used in particular for studying orbit spaces for actions on contractible spaces. When G is a compact Lie group acting freely on X, then n :X/H~X/G is a fiber bundle with fiber G/H, and the transfer constructed by Becker and Gottlieb [1] applies in this case. So our goal here is to generalize both of these constructions. One obvious consequence of the above theorem is that