A COMPACT IMBEDDING OF RIEMANNIAN SYMMETRIC SPACES

Let G be a connected real semisimple Lie group with finite center and  be a Cartan involution of . G Suppose that K is the maximal compact subgroup of G corresponding to the Cartan involution .  The coset space = / G K X is then a Riemannian symmetric space. In this paper, by choosing reduced root system = { | 2 ; / 2 }         instead of restricted root system  and using the action of the Weyl group, first we construct a compact real analytic manifold ' X in which the Riemannian symmetric space / G K is realized as an open subset and that G acts analytically on it; then, we consider the real analytic structure of ' X induced from the real analytic structure of IR , A the compactification of the corresponding vectorial part.


Introduction
Let G be a connected real semisimple Lie group with finite center and g be the Lie algebra of G .Denote the Cartan involution of G by  and K the fixed points of . Then, K is a maximal compact subgroup of G , and the coset space =/ GK X becomes a Riemannian symmetric space.We also denote by  the Cartan involution of g , corresponding to the Cartan involution  of .
G It follows that =  g k p is the Cartan decomposition of g into eigenspaces of ,  where k is the Lie algebra of K .Let a be a maximal abelian subspace in p and * a be the dual space of .a The corresponding analytic subgroup A of a in G is, then, called the vectorial part of .X For a non zero *  a , non zero eigenspace g Moreover, Weyl group W of  is defined with normalizer () K It acts naturally on a and coincides via this action with the reflection group of the root system . Choose a fundamental system 1 = { ,..., } l   of ,  where number l , which equals dim a , is called the split rank of the symmetric space X and denote the corresponding set of all restricted positive roots in  by .GK in a compact real analytic manifold.In [3] , we applied the construction for semisimple symmetric spaces, and determined the system of invariant differential operators on the corresponding compactifications [4].
In this paper, by choosing reduced root system and using the action of the Weyl group, first we construct a compact real analytic manifold ' X in which the Riemannian symmetric space / GK is realized as an open subset and that G acts analytically on it; then, we consider the real analytic structure of ' X induced from the real analytic structure of IR .A Our construction is a motivation for the construction of Oshima and Sekiguchi [7] for affine symmetric spaces and it is similar to those in [6], [8], [9] for semisimple symmetric spaces.

A compactification of the vectorial part
In this section, we recall some notations and results concerning compactification [2] and then illustrate the construction via the case of symmetric space ( , IR) / ( ).

SL n
SO n Let G be a connected real semisimple Lie group with a finite center and g be the Lie .
Denote the 1-dimensional complex projective space by Then, we get a maximal abelian subspace in g defined by defines an atlas of charts on M such that compact manifold M is covered by !n -many charts.By definition, we see that Then, we get homeomorphism is a compact real analytic manifold and set of charts defines an atlas of charts on IR A so that manifold

IR
A is covered by !n -many charts.

A realization of Riemannian symmetric spaces
Consider subset  is defined by is an extended signature of roots that is defined in [ FF       Then, (see [8]) following subsets

XX gives the orbital decomposition of ' X
for the action of G on it. (ii) is connected for every wW   , and W  is generated by elements  is a fundamental system of roots for ,   we see that the quotient space ' X is connected.Consider compact subset is also compact because it is the image of a compact set under continuous map .


Let  a denote the positive chamber corresponding to   and put a is a fundamental domain for the action of , x W so we can apply Lemma 2.5 in [7] to imply that compact set X Then, ' X is also compact.
(ii) Put  is the composition of the above maps It follows from what we have mentioned at the beginning of the proof, maps   is an open covering of ' X such that maps w g  are real analytic diffeomorphisms.
(ii) The action of G on ' X is analytic and orbit ()  is an open covering of '.

X
a is called the root space, and the corresponding s   the restricted root.Then, the set system with the inner product induced by the Killing form <, > of .

A
], based on the natural embedding of * () into a compact real analytic manifoldIR A which is called a compactification of IR ,A and then constructed a realization of / are analytic diffeomorphisms between the open subsets of IR .N  Moreover, by a similar way as the proof of Lemma 2.8 (iii) in[7], we can show that the map

 3 . 5
also has the same property, we have the Lemma.Lemma 3.3 and Lemma 3.4 assure that we can define an analytic structure on ' so that they define analytic diffeomorphisms onto open subsets w g  of ' X and the action of G on ' X is analytic.On the other hand, based on the homeomorphism between gN    and || IR gN  for every gG  and by a similar way as the proof of Theorem 2.7 in[7], we can prove that topological space ' X is Hausdorff.Moreover, for an element , wW   the unique G -orbit which is isomorphic to / GK (resp.Combining this with Proposition 3.2, we get Theorem The quotient topological space ' X has the following properties: (i) ' X is a compact connected real analytic manifold and , algebra of G .Denote the complexification of g by 7, Definition 2.1].Now, we go to define parabolic subalgebras with respect to extended signatures ( ), a we go to define an equivalent relation for points in In particular, the number of G -orbits which are isomorphic to /