Zeta functions and their asymptotic expansions for compact symmetric spaces of rank one

Let M be a compact Riemannian symmetric space. Then M=G/K, where G is the identity component of the isometry group of M and K is the isotropy subgroup of G at a point. In 1965 Nagano studied and classified the geometric transformation groups of compact symmetric spaces. Roughly speaking they are ‘larger’ groups L that act on M, (i) G/L; (ii) L is a Lie transformation group acting effectively on M; (iii) L preserves the symmetric structure of M; and (iv) L is simple. Using ‘Helgason spheres’, S(α), the minimal totally geodesic spheres in a compact irreducible symmetric space, we define an arithmetic distance for compact irreducible symmetric spaces and prove: THEOREM. Let M=G p(K n ), K=ℂ, H, or R, or M=AI(n), of rank greater that 1 and dimension greater that 3, let L′ be the geometric transformation group of M. Let L={ϕ: M→M: ϕ is a diffeomorphism and ϕ preserves arithmetic distance}. Then L=L′

We determine the stabilityof totally geodesic submanifolds in a compact symmetric space, which are called polars and meridians (see 2.1). These subspaces were introduced by Chen and Nagano ([CN-1]) and we have many interesting results after that ([CN-2], [N-l], [N-2], [NS-1], [NS-2] and [NS-3]). Recently, several results have been obtained about the stabilityof totally geodesic submanifolds in compact symmetric spaces. Ohnita gave the formula for the index, the nullityand the Killing nullity of a totally geodesic submanifold in a compact symmetric space in [0], in which he also proved that the Helgason sphere in a compact symmetric space is stable. Tasaki proved that the Helgason sphere in a compact Lie group is homologically volume-minimizing in its real homology classin [Ts-1]. He used the calibrationtheory. And there are studies about the stability of certain closed subgroups in a compact Lie group by Mashimo and Tasaki ([MT-1] and [MT-2]). Mashimo determined all the unstable Cartan embeddings of compact symmetric spaces in [M]. And there is a result about the stabilityof symmetric i?-spaces in Hermitian symmetric spaces and totally complex submanifolds in quaternionic Kahler symmetric spaces of classical type by Takeuchi ([Tk-2]). Recently Nagano and the author have obtained a result on a relationshipbetween the stabilityof minimal submanifolds and that of /^-harmonic maps ([NS-3]). In the present paper we study the stability of all the polars and meridians in every compact symmetric space by using Ohnita's method in Section 3. We will also study the stabilityof totally complex totallygeodesic submanifolds in quaternionic Kahler symmetric spaces of exceptional type in Section 4. I should like to express my gratitude to Professor T. Nagano for his useful advice and kindly support. This is the author's doctoral dissertation submitted to Sophia University in October, 1992.

Introduction. In this paper we apply the theory of Bott and Samelson [6] to the study of the integral singular homology of certain spaces of paths on a compact space. It should be noted that a special case of one of our theorems (Theorem 2.1) has been obtained independently by S. Araki and will appear in his forthcoming study of the Bott-Samelson K-cycles associated to spaces [2]. Some of the technical lemmas involved in the proof of Theorem 2.2 are also known to Araki, but are used by him for quite different purposes. The spaces of paths to be considered are those associated to the variationally complete group actions described by Hermann in [7]. Thus if G is a compact connected Lie group and K, H are subgroups (for our purposes a subgroup is the full fixed point group of an involutive automorphism of G) then we let N be a K-orbit in G/ H and set = Q(G/ H; x, N), the space of paths in G/ H which start at the point x and terminate on N. Our principal results will be a complete determination of the K-cycles in Q2 (cf. [6, pp. 969-972]) and, as an application of this, the formulation of a necessary and, sufficient condition that the singular homology H*(Q) be free of torsion. Any effective application of the Bott-Samelson theory to Q2 will require a description of the distribution and defects of the K-orbits in G/ H. For the case K = H it is known (cf. [6]) that such a description is provided by a maximal torus of G/ K together with a certain diagram of singular subtori determined by the root system of G/K. We generalize this situation to the symmetric triad (G; K, H) by selecting a geodesically imbedded torus Tin G/ H which meets all the K-orbits and meets them orthogonally (cf. [8]), the singular points of which again fall into a finite union of subtori. Some propositions of Siebenthal proven in [12] then provide the key to a complete description of these singular

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Minimal symmetric submani folds in compact Riemannian symmetric spaces

A relatively simple algebraic framework is given, in which all the compact symmetric spaces can be described and handled without distinguishing cases. We also give some applications and further results.

This chapter is devoted to so-called “freak theorems” and their local analogues. Results due to Radon, Ungar, Schneider, Berenstein, and Zalcman are described. These results are extended to convolution equations on arbitrary compact symmetric spaces. For compact symmetric spaces of rank one, local analogues of “freak theorems” are obtained. It is shown that all the assumptions in these results cannot be relaxed.

Paley–Wiener type theorems describe the image of a given space of functions, often compactly supported functions, under an integral transform, usually a Fourier transform on a group or homogeneous space. In this article we proved a Paley–Wiener theorem for smooth sections f of homogeneous line bundles on a compact Riemannian symmetric space . It characterizes f with small support in terms of holomorphic extendability and exponential growth of their χ‐spherical Fourier transforms, where χ is a character of K. An important tool in our proof is a generalization of Opdam's estimate for the hypergeometric functions associated to multiplicity functions that are not necessarily positive. At the same time the radius of the domain where this estimate is valid is increased. This is done in an appendix.

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On the Analytic Continuation of the Minakshisundaram–Pleijel Zeta Function for Compact Symmetric Spaces of Rank One

A new characterization is provided for the class of compact rank-one symmetric spaces. Such spaces are the only symmetric spaces of compact type for which the standard vector field xi ^{S} on their sphere bundles is Killing with respect to some invariant Riemannian metric. The set of all these metrics is determined, as well as the set of all those invariant contact metric structures with characteristic vector field xi ^{S}. Moreover, on tangent sphere bundles of compact symmetric spaces with rank greater than or equal to two, a family of invariant contact metric structures, which contains the standard structure, is obtained.

We define two types of Witten’s zeta functions according to Cartan’s classification of compact symmetric spaces. The type II is the original Witten zeta function constructed by means of irreducible representations of the simple compact Lie group U. The type I Witten zeta functions, we introduce here, are related to the irreducible spherical representations of U. They arise in the harmonic analysis on compact symmetric spaces of the form U/K, where K is the maximal subgroup of U. To construct the type I zeta function we calculate the partition functions of 2d YM theory with broken gauge symmetry using the Migdal–Witten approach. We prove that for the rank one symmetric spaces the generating series for the values of the type I functions with integer arguments can be defined in terms of the generating series of the Riemann zeta-function.

The concept of t-designs in compact symmetric spaces of rank 1 is a generalization of the theory of classical t-designs. In this paper we obtain new lower bounds on the cardinality of designs in projective compact symmetric spaces of rank 1. With one exception our bounds are the first improvements of the classical bounds by more than one. We use the linear programming technique and follow the approach we have proposed for spherical codes and designs. Some examples are shown and compared with the classical bounds.

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Harmonic analysis and propagators on homogeneous spaces