Volume 21, Issue 3

In the finite-dimensional setting, every Hermitian-symmetric space of compact type is a coadjoint orbit of a finite-dimensional Lie group. It is natural to ask whether every infinite-dimensional Hermitian-symmetric space of compact type, which is a particular example of an Hilbert manifold, is transitively acted upon by a Hilbert Lie group of isometries. In this paper we give the classification of infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits of connected simple L *-groups of compact type using the notion of simple roots of non-compact type. The key step is, given an infinite-dimensional symmetric pair (, ), where is a simple L *-algebra of compact type and a subalgebra of , to construct an increasing sequence of finite-dimensional subalgebras n of together with an increasing sequence of finite-dimensional subalgebras n of such that , , and such that the pairs ( n , n ) are symmetric. Comparing with the classification of Hermitian-symmetric spaces given by W. Kaup, it follows that any Hermitian-symmetric space of compact or non-compact type is an affine-coadjoint orbit of an Hilbert Lie group.

Let X = G / K be a Riemannian symmetric space and ⊂ D G ( X ) a co-finite ideal. A function F on X is -harmonic if F = 0. A result of Oshima and Sekiguchi [Advanced Studies in Pure Math. 4: 391–432, 1984] says such a function is the Poisson integral of a distribution over the Furstenberg boundary G / P if and only if it has moderate growth. We prove a partial generalization of this result to general, non-symmetric, bounded homogeneousdomains in ℂ n . Instead of D G ( X ), we use an algebra of geometrically defined differential operators, D geo ( X ) which, in the symmetric case, is a subalgebra of D G ( X ). We prove the existence of an asymptotic expansion for -harmonic functions that reduces in the symmetric space case to the expansions due to Wallach [Lie algebra cohomology and holomorphic continuation of generalized Jacquet integrals, Academic Press, 1988] and van den Ban and Schlichtkrull [J. Reine Angew. Math. 380: 108–165, 1987]. We prove a convergence theorem for these expansions that seems to be new even in the symmetric space case. These expansions are used to define boundary values for F which uniquely determine F . An algorithm constructing F from its boundary values is given.

Let R be a commutative ring, G a group and RG its group ring. Let φ σ : RG → RG denote the involution defined by φ σ (Σ r g g ) = Σ r g σ( g ) g –1 , where σ : G → {±1} is a group homomorphism (called an orientation morphism). An element x in RG is said to be antisymmetric if φ σ ( x ) = – x . We give a full characterization of the groups G and its orientations for which the antisymmetric elements of RG commute.

On compact Riemannian manifolds ( M , g ) of dimension n ≥ 2 with smooth boundary, the gradient estimates for the eigenfunctions of the Dirichlet Laplacian are proved by the maximum principle. Using the L ∞ estimates and gradient estimates, the Hörmander multiplier theorem is shown for the eigenfunction expansion of Dirichlet Laplacian on compact manifolds with boundary.

We prove that a loop Q has to be conjugacy closed modulo Z ( Q ) whenever all mappings L ( x , y ) and R ( x , y ) are automorphisms, N ( Q ) ⊴ Q , Q / N ( Q ) is an abelian group, and 〈 L x ; x ∈ Q 〉 is a normal subgroup of the multiplication group.

Let ƒ be a self-map of a smooth compact connected and simply-connected manifold of dimension m ≥ 3, r a fixed natural number. In this paper we define a topological invariant which is the best lower bound for the number of r -periodic points for all C 1 maps homotopic to ƒ. In case m = 3 we give the formula for and calculate it for self-maps of S 2 × I .

It is proved here that a function in ℝ 2 which is separately real analytic in one variable and CR extendible in the other (that is separately holomorphically extendible to a uniform strip), is real analytic. It is also considered the case when the CR extendibility occurs only on one side. The proof is obtained by bringing the problem into the frame of CR geometry.

A well-known result due to H. Corson has been recently improved by the authors. In its final form it essentially reads as follows: for any covering τ by closed bounded convex subsets of any Banach space X containing a separable infinite-dimensional dual space, a (algebraically) finite-dimensional compact set C can always be found that meets infinitely many members of τ . In the present paper we investigate how small the dimension of this compact set can be, in the case the members of τ are closed bounded convex bodies satisfying general conditions of rotundity or smoothness type. In particular, such a compact set turns out to be a segment whenever the members of τ are rotund or smooth bodies in the usual sense.

We continue our study of genus 2 curves C that admit a cover C → E to a genus 1 curve E of prime degree n . These curves C form an irreducible 2-dimensional subvariety ℒ n of the moduli space ℳ 2 of genus 2 curves. Here we study the case n = 5. This extends earlier work for degree 2 and 3, aimed at illuminating the theory for general n . We compute a normal form for the curves in the locus ℒ 5 and its three distinguished subloci. Further, we compute the equation of the elliptic subcover in all cases, give a birational parametrization of the subloci of ℒ 5 as subvarieties of ℳ 2 and classify all curves in these loci which have extra automorphisms.