Some results on Semisimple Symmetric Spaces and Invariant Differential Operators

<pre>Let X = G/H be a semisimple symmetric space of non-compact style. Our purpose is to construct a compact real analytic manifold in which the semisimple symmetric space X = G/H is realized as an open subset and that $G$ acts analytically on it.</pre><pre> By the <span>Cartan</span> decomposition <span>G = KAH,</span> we must <span>compacify</span> the <span>vectorial</span> part <span>A.$</span></pre><pre> In [6], by using the action of the Weyl group, we constructed a compact real analytic manifold in which the semisimple symmetric space G/H is realized as an open subset and that G acts analytically on it.</pre><pre>Our construction is a motivation of the <span>Oshima's</span> construction and it is similar to those in N. <span>Shimeno</span>, J. <span>Sekiguchi</span> for <span>semismple</span> symmetric spaces.</pre><pre>In this note, first we will <span>inllustrate</span> the construction via the case of <span>SL (n, </span>R)/SO_e (1, n-1) and then show that the system of invariant differential operators on X = G/H extends analytically on the corresponding compactification. </pre>

On the Geometry of the Moduli Space of Real Binary Octics

The moduli space of smooth real binary octics has five connected components. They para- metrize the real binary octics whose defining equations have 0, … , 4 complex-conjugate pairs of roots respectively. We show that each of these five components has a real hyperbolic structure in the sense that each is isomorphic as a real-analytic manifold to the quotient of an open dense subset of 5- dimensional real hyperbolic space by the action of an arithmetic subgroup of Isom. These subgroups are commensurable to discrete hyperbolic reflection groups, and the Vinberg diagrams of the latter are computed.

Complexification and Hypercomplexification of Manifolds with a Linear Connection

We give a simple interpretation of the adapted complex structure of Lempert–Szöke and Guillemin–Stenzel: it is given by a polar decomposition of the complexified manifold. We then give a twistorial construction of an SO(3)-invariant hypercomplex structure on a neighbourhood of X in TTX, where X is a real-analytic manifold equipped with a linear connection. We show that the Nahm equations arise naturally in this context: for a connection with zero curvature and arbitrary torsion, the real sections of the twistor space can be obtained by solving Nahm's equations in the Lie algebra of certain vector fields. Finally, we show that, if we start with a metric connection, then our construction yields an SO(3)-invariant hyperkähler metric.

A Property of Lie Group Orbits

Let G be a real Lie group and X a real analytic manifold. Suppose that G acts analytically on X with finitely many orbits. Then the orbits are subanalytic in X. As a consequence we show that the micro-support of a G-equivariant sheaf on X is contained in the conormal variety of the G-action.

Explicit invariant solutions for invariant linear differential operators

SynopsisLet F be a real analytic function on a real analytic manifold X. Let P be a linear differential operator on X such that , where Q is an ordinary differential operator with analytic coefficients whose singular points are all regular. For each (isolated) critical value z of F, we construct locally an F-invariant solution u of the equation Pu - v, v being an arbitrary F-invariant distribution supported in F−1(z). The solution u is constructed explicitly in the form of a series of F-invariant distributions.

Deformations of real analytic functions and the natural stratification of the space of real analytic functions

Let A be a real analytic set, M be a compact real analytic manifold and f : A × M → R be a real analytic function. Then we have a family of real analytic functions fa, a ∈ A, on M defined by fa(X) = f(a, x).