Symmetric Spaces Of Compact Type Research Articles

Convergence of Fourier series on compact Lie groups

The main result is an LP mean convergence theorem for the partial sums of the Fourier series of a class function on a compact semisimple Lie group. A central element in the proof is a Lie group-Lie algebra analog of the theorems in classical Fourier analysis that allow one to pass back and forth between multiplier operators for Fourier series in several variables and multiplier operators for the Fourier transform in Euclidean space. To obtain the LP mean convergence theorem, the theory of the Hilbert transform with weight function is needed. Introduction. A theorem of M. Riesz says that if f is in LP of the circle, 1 < p < oo, and if SNf(x)__Nakeikx is the Nth partial sum of the Fourier series of f, then SN f converges to f in the LP norm as N -* . Pollard [15] proved a similar result for Jacobi polynomials on the interval [1, 1]. If fis inLP([-1, 1]; (1 -x)'(l -x)dx) and if N SN f (X) =E do ', Oa Rk ' (X) k=O is the Nth partial sum of the Jacobi series of f, then SN f converges to f in the LP norm provided 4 ma+ ? 1< <4min o+l 3 +1 Xe+ '2,B + 3, 2 F1 + 1 Here R'' 0 is a normalized Jacobi polynomial and dk' p is an appropriate constant. It is well known that for suitable choices of a, 3 the {R 'kn} are the elementary spherical functions for the rank 1 symmetric spaces of compact type. Cast in this setting Pollard's theorem is an LP mean convergence result for bi-K invariant functions on the rank 1 compact symmetric space U/K. In this paper we investigate extending this result to higher rank symmetric spaces. Even in the abelian case of the n-torus Tn the result depends drastically on how the multiple series is summed. Consider for example f in LP(Tn) with Received by the editors October 3, 1974 and, in revised form, March 12, 1975. AMS (MOS) subject classifications (1970). Primary 43A90, 43A75; Secondary 33A45, 33A75. 61 Copyright @ 1976, American Mathematical Society This content downloaded from 207.46.13.58 on Sat, 17 Sep 2016 05:41:43 UTC All use subject to http://about.jstor.org/terms

Totally real representations and real function spaces

0* Introduction* Let X be a riemannian symmetric space of compact type and G its largest connected group of isometries. In his 1929 paper [1] on class 1 representation s, E. Cartan showed that the symmetry of X sends every uniformly closed G-invariant function space on X to its complex conjugate. Starting from the point of view of algebras, Mirkil and de Leeuw [4] showed that every rotation invariant function algebra on the sphere Sn(n ^ 2) was spanned by real-valued functions, hence (Stone-Weierstrass theorem) that such an algebra necessarily was all continuous functions on Sn, all continuous functions on real protective w-space, or just the constantsthat state of affairs is quite different from the case n = 1. When the rotation group S0(n + 1) contains the symmetry of S w, i.e. when n is even, Cartan's result mentioned above implies reality of such function algebras. The published Mirkil-de Leeuw argument rests rather on the fact that the spherical harmonics are real-valued.

On Non-existenceness of Equifocal Submanifolds with Non-flat Section

We first prove a certain kind of splitting theorem for an equifocal submanifold with non-flat section in a simply connected symmetric space of compact type, where an equifocal submanifold means a submanifold with parallel focal structure. By using the splitting theorem, we prove that there exists no equifocal submanifold with non-flat section in an irreducible simply connected symmetric space of compact type whose codimension is greater than the maximum of the multiplicities of roots of the symmetric space or the maximum added one. In particular, it follows that there exists no equifocal submanifold with non-flat section in some irreducible simply connected symmetric spaces of compact type and that there exists no equifocal submanifold with non-flat section in simply connected compact simple Lie group whose codimension is greater than two.

Collapse of the Mean Curvature Flow for Equifocal Submanifolds

In this paper, we investigate the mean curvature flows for an equifocal submanifold in a symmetric space of compact type and its focal submanifolds as initial data. It is known that equifocal submanifolds of codimension greater than one in irreducible symmetric spaces of compact type occur as principal orbits of Hermann actions. However, we investigate the flows conceptionally without use of this fact. Concretely the investigation is performed by investigating the mean curvature flows having the lifts of the submanifolds to an (infinite deiminsional separable) Hilbert space through a Riemannian submersion as initial data.