SOME RESULTS ON SEMISIMPLE SYMMETRIC SPACES AND INVARIANT DIFFERENTIAL OPERATORS

: Let 𝐺 be a connected real semisimple Lie group with finite a center and 𝜎 be an involutive automorphism of 𝐺. Suppose that 𝐻 is a closed subgroup of 𝐺 with 𝐺 𝑒𝜎 ⊂ 𝐻 ⊂ 𝐺 𝜎 , where 𝐺 𝜎 is the fixed points group of 𝜎 , and 𝐺 𝑒𝜎 denotes its identity component. The coset space 𝐗 = 𝐺/𝐻 is then a semisimple symmetric space. Our purpose is to construct a compact real analytic manifold 𝐗̂ in which the semisimple symmetric space 𝐗 = 𝐺/𝐻 is realized as an open subset, and that 𝐺 acts analytically on it. Using the Cartan decomposition 𝐺 = 𝐾𝐴𝐻, we must compactify the vectorial part 𝐴. In [6], 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. Our construction is a motivation of the Oshima’s construction, and it is similar to those in Shimeno and Sekiguchi for semisimple symmetric spaces. In this note, first we illustrate the construction via the case of 𝑆𝐿(𝑛,IR)/𝑆𝑂 𝑒 (1,𝑛 − 1) and then show that the system of invariant differential operators on 𝐗 = 𝐺/𝐻 extends analytically on the compactification 𝐗̂.


Introduction
Let be a connected real semisimple Lie group with a finite center, be an involutive automorphism of and = / be the corresponding semisimple symmetric space. Here is a closed subgroup of with ⊂ ⊂ , where is the fixed points group of , and denotes its identity component.
Denote for the Cartan involution which commutes with and the fixed points of . Then, is a -stable maximal compact subgroup of . Let be the Lie algebra of . The involutions of induced by and are denoted by the same letters, respectively.
Suppose that = ⊕ = ⊕ is the decompositions of into +1 and -1 eigenspaces for and , respectively, where (resp. ) is the Lie algebra of (resp. ). Fix a maximal abelian subspace in ∩ and let * denote the dual space of . The corresponding analytic subgroup of in is then called the vectorial part of . For a ∈ * , put = { ∈ | [ , ] = ( ) , ∀ ∈ }.
Then the set Σ = { ∈ * | ≠ {0}, ≠ 0} defines a root system with the inner product induced by the Killing form <, > of . Moreover, the Weyl group of Σ is defined with the normalizer ( ) of in , modulo the centralizer = ( ) of in . It acts naturally on and coincides via this action with the reflection group of the root system Σ.
Let Δ = { 1 , . . . , } be a fundamental system of Σ, where the number which equals dim is called the split rank of the symmetric space , and denote Σ + for the corresponding set of all positive roots in Σ , and ∩ for the normalizer ∩ ( ) of in modulo the centralizer ∩ ( ) of in . We see that ∩ is a subgroup of . For each element of we fix a representative in ( ) so that ∈ ∩ ( ) if ∈ ∩ . In [6], using the action of the Weyl group, we constructed a compact real analytic manifold in which the semisimple symmetric space / is realized as an open subset, and that acts analytically on it. By this way, we first, construct an imbedding of IR into a compact real analytic manifold ̂I R which is called a compactification of IR . Then, we construct the compact manifold ̂ based on the action of Weyl group on ̂I R and consider the real analytic structure of ̂ induced from the real analytic structure of ̂I R . Our construction is a motivation of the Oshima's construction, and it is similar to those in Shimeno and Sekiguchi for semisimple symmetric spaces.
In this note, first, we recall some notation and results concerning the compactification of semisimple symmetric spaces constructed in [6] and illustrate the construction via the case of ( , IR)/ (1, − 1). Then, we show that the system of invariant differential operators on = / extends analytically on the compactification ̂.

A compact imbedding of symmetric spaces
In this section, we recall some notation and results concerning the compactification of semisimple symmetric spaces constructed in [6].

2.2
Using the natural imbedding of ( * ) Σ into ( IP 1 ) Σ , we get an imbedding map of into ( IP 1 ) Σ denoted also by .
By definition, ( IR ) is a subset of (IRIP 1 ) Σ . Denote ̂I R for the closure of ( IR ) in (IRIP 1 ) Σ . It follows from (2.1) and (2.2) that ̂I R is a compact subset in . Now, we define an atlas of charts on ̂I R induced from the atlas on .

Theorem 2.1 ̂ is a compact real analytic manifold that is called a compactification of
. The set of charts { ( ) , ( ) } ∈ defines an atlas of charts on ̂ so that the manifold ̂ is covered by | |-many charts.
First, we consider ̃∈̂I R − , and let = { | = (̃) ≠ 0 } be a subset of the simple root system Δ with respect to the extended signature = (̃). Denote Σ = (∑ ∈ IR ) ∩ Σ and suppose that is the subgroup of generated by reflections with respect to in . Let be the parabolic subgroup of with the corresponding Langlands decomposition = so that is the centralizer of in , and the Lie algebra of equals ∑ ∈Σ + . Then, (see [11]) we define a parabolic subalgebra of and its Langlands decomposition = + + so that ⊂ .
Using , Σ and instead of , Σ and , respectively, we define a parabolic subalgebra Then, Definition 2.2 really gives an equivalence relation, which we write ~′. The quotient space of ×̂I R by this equivalence relation becomes a topological space with the quotient topology and denoted by ̂.
Let : ×̂I R →̂ be the natural projection. As the action of on ×̂I R is compatible with the equivalence relation, we can define an action of on ̂ by   It follows that the pair { Σ + , Σ + } is a chart on , and { (Σ + ) , (Σ + ) } ∈ defines an atlas of charts on such that the manifold is covered by !-many charts. By definition, we see that Then, we get a homeomorphism Δ : Δ → IR −1 defined by Δ ( ) = ( ) ∈Δ , ∀ ∈ Δ .
Hence, ̂I R ≅ IR −1 ∪ {∞} ≅ −1 is a compact real analytic manifold, and the set of charts { (Δ) , (Δ) } ∈ defines an atlas of charts on ̂I R so that the manifold ̂I R is covered by !many charts.
It follows from Proposition 2.3 that the compactification ̂ of the symmetric space = ( , IR)/ (1, − 1) is a compact connected ( , IR)-space, and there are just 2 −1 open orbits that are isomorphic to ( , IR)/ (1, − 1). Moreover, the number of compact orbits in ̂ equals , the number of elements of the coset ∩ \ .

Invariant differential operators
In this section, we shall show that the system of invariant differential operators on = / extends analytically on the compact -space ̂. First, we recall after [9] on the structure of the algebra of invariant differential operators on / .
For a real or complex Lie subalgebra of , we denote ( ) for the universal enveloping algebra of ′, where ′ is the complex Lie subalgebra of , generated by . Now, we retain the notation in section 2. First, the complex linear extensions of the involution and on are also denoted by the same letters. Let be a maximal abelian subspace of containing . Denote Σ( ) for the root system for the pair ( , ) and Σ( ) + for the set of positive roots with respect to the compatible orders for Σ( ) and Σ. Put = Let be the projection of ( ) to ( ) with respect to this decomposition and be the algebra automorphism of ( ) defined by ( ) = − ( ) for ∈ . Then, the map = ∘ induces the Harish-Chandra isomorphism: Denote ( / ) for the algebra of invariant differential operators on / and ( / 0 ) for the algebra of invariant differential operators on / 0 . Then, we see that ( / 0 ) is naturally isomorphic to the algebra ( ) /( ( ) ∩ ( ) ), and it follows from Lemma 4.1 that ( / ) is also isomorphic to this algebra. Hence, we get the algebra isomorphism by identifying algebras ( / ) and ( ) /( ( ) ∩ ( ) ). Moreover, we have the natural projection which satisfies = ∘ .
Now, we will study the -invariant differential operators on the -manifold ̂ constructed in Section 2 based on the invariant differential operators on the manifold = / .
Denote (̂) for the algebra of -invariant differential operators on the manifold ̂ whose coefficients are real analytic functions. Then, we have
Since is open dense in ̂, in order to get the theorem we have only to show that has an analytic extension on ̂. From Theorem 3.6 in [6], we get that ̂= ⋃ ∈ ,1≤ ≤ Ω .
Based on this, we see that has an analytic extension on ̂ if | ∩ Ω has an analytic extension on Ω . By a similar way with the proof of Theorem 2.5 in [9], we can prove this assertion. Accordingly, the theorem is proved.