In this paper, we shall study the real nilpotent orbits of vector space associated to a semisimple symmetric pair.Let $\mathfrak{g}$ be Lie algebra and let $\sigma$ its involution.Then obtain direct sum decomposition $\mathfrak{g}=\mathfrak{h}+q$ for .The pair $(\mathfrak{g}, \mathfrak{h})$ is called pair.The first purpose paper prove theorem concerning H-orbital structure subvariety $\mathfrak{N}(q)$ $q$ second determine orbital when split rank one in sense [OS, Def. 2.5.1].We are going...

(1980). On a certain generator system of the ring invariants finite reflection group. Communications in Algebra: Vol. 8, No. 4, pp. 373-408.

Let G be a real semisimple Lie group with finite center, and K maximal compact subgroup of G. A zonal spherical function on the symmetric space X=G/K is an simulatneous eigeiifunction (p(x) all invariant differential operators X satisfying (p(kx) = cp(x) for any x^X, k^K, (p(eK) =1, where e identity element in By Cartan decomposition KAK, considered as A. And by separation variables, we obtain from operators, which are called their radial components. In this paper, investigate components...

Flat structure was introduced by K. Saito and his collaborators at the end of 1970's. Independently WDVV equation arose from 2D topological field theory. B. Dubrovin unified these two notions as Frobenius manifold structure. In this paper, we study isomonodromic deformations an Okubo system, which is a special kind systems linear differential equations. We show that space independent variables such can be equipped with (without metric), C. Sabbah generalization manifold. As its consequence,...

Ever since S. Helgason [4] showed that any eigenfunction of the Laplace-Beltrami operator on unit disk is represented by Poisson integral a hyperfunction circle, much interest has been arisen to study representation joint eigenfunctions all invariant differential operators symmetric space X. In particular, his original idea expanding into K -finite functions proved be generalizable up case where X Riemannian rank one (cf. [4], [5], [11]). Presently, extension arbitrary completed quite...

For the notation, see [9].Let $X$ be an l-dimensional complex mani- fold and let $T^{*}X$ cotangent bundle over .Let, further, $\pi$ denote projection of to .A holonomic system $\mathscr{L}$ is then a coherent $\mathcal{D}_{X}$ -Module with $co\dim_{TX}SS(\mathscr{L})=l\vee$ .An irreducible component $ SS(\mathscr{L})\vee$ called variety.The holonomy diagram determined as follows.To each variety $\Lambda$ corresponds circle inscribing .The \copyright -\copyright means that two varieties...

We describe an approach to classification of weighted homogeneous Saito free divisors in C 3 . This is mainly based on properties Lie algebras vector fields tangent reduced hypersurfaces at their non-singular points. In fact we also obtain a such having similar as ones for discriminants associated with irreducible real reflection groups rank 3. Among other things briefly discuss some applications the theory 3, interesting relationships root systems types E 6 , 7 8 and few examples higher...

leave $f(x)=f_{W}(x)$ invariant.More precisely, we havewith certain polynomials $c_{i}(x)\in R$ .Furthermore, $X_{1},$ $\cdots,$ $X_{l}$ form a free basis of the Lie algebra vector fields leaving set $\{x;f(x)=0\}$ invariant ([7]).In this paper, shall study microlocal structure $\mathcal{D}_{X}$ -Module

Let l1,l2,...,l7 be mutually different seven lines on the real projective plane. We consider two conditions;(A) No three of intersect at a point. (B) There is no conic tangent to any six l1,l2, . , l7. Cummings [3] and White [16] showed that there are eleven non-equivalent classes systems with condition (A)(cf. [7], Chap. 18). The purposes this article give an interpretation classification in terms root system type E7. To accomplish this, it better add for lines. Moreover we need notion...

Let ${\mathcal O}$ be a nilpotent orbit in $\mathfrak{so}(p,q)$ under the adjoint action of full orthogonal group ${\rm O}(p,q)$. Then closure (with respect to Euclidean topology) is union and some O}(p,q)$-orbits smaller dimensions. In an earlier work, first author has determined which belong this closure. The same problem for identity component SO}(p,q)^0$ O}(p,q)$ on much harder we propose conjecture describing closures SO}(p,q)^0$-orbits. proved when $\min(p,q)\le7$. Our method indirect...

0. Introduction.Let g0 be a real semisimple Lie algebra and let 0+0 Cartan decomposition of go.We complexify 0, 0 P0 denote them by g, p, respectively.Let G the adjoint group K analytic subgroup corresponding to

A tritangent plane., or a for short, means projective plane which meets the surface in union of three lines.Since exactly 5 tritangents pass through each lines upon surface, it follows that famous theorem: There are 27 the.general cubic surface.

A potential vector field is a solution of an extended WDVV equation which generalization equation.It expected that fields corresponding to algebraic solutions Painlevé VI can be written by using polynomials or functions explicitly.The purpose this paper construct more than thirty non-equivalent solutions.

<!-- *** Custom HTML --> The aim of this report is to determine the fundamental group an arbitrary irreducible semisimple symmetric space $G/H$ when $G$ a connected Lie with trivial center. $\pi_1(G/H)$ well-known if Riemannian. Therefore, we restrict our attention case where non-Riemannian so both and $H$ are not compact. result summarized in Table 4.