# Algebraic Structures of Symmetric Domains

Ichiro Satake
Pages: 342
https://www.jstor.org/stable/j.ctt7ztrxs

1. Front Matter
(pp. i-vi)
2. Preface
(pp. vii-viii)
I. Satake
(pp. ix-x)
(pp. xi-xiv)
5. List of symbols
(pp. xv-xvi)
6. Chapter I Algebraic Preliminaries
(pp. 1-43)

LetFbe a field of characteristic zero and letVbe a vector space of dimensionnoverF. We denote by End(V) the algebra of allF-linear transformations ofVand byGL(V) the group of units in End(V), i. e., the group of all non-singularF-linear transformations ofV. When a basis ofVis fixed,Vis identified with the space ofn-tuplesFn, and so End(V) andGL(V) are also identified with ℳn(F), the full matrix algebra of degreenoverF, andGLn(F), the general linear group of degreenoverF, respectively.

In...

7. Chapter II Basic Concepts on Symmetric Domains
(pp. 44-88)

In this section, we summarize some basic concepts and results on (globally) symmetric Riemannian manifolds. The main references will be Kobayashi-Nomizu [1], abbreviated as K-N, and Helgason [1], [la]. All manifolds and mappings to be considered in this chapter are supposed to beC.

LetMbe a connected Riemannian manifold with a (C) Riemannian metricq. We denote byI(M) the group of isometries ofMand byI(M)xthe isotropy subgroup atxϵM(i. e., the stabilizer ofxinI(M)). It is known (K-N, VI, Th. 3. 4) thatI(M) with the compact-open topology becomes...

8. Chapter III Unbounded Realizations of Symmetric Domains (Theory of Wolf-Korányi)
(pp. 89-164)

Let$(\mathfrak{g},{{H}_{0}})$and$({{\mathfrak{g}}^{\prime }},H_{0}^{\prime })$be two semi-simple Lie algebras of hermitian type. We recall that a homomorphism$\rho :\mathfrak{g}\to {{\mathfrak{g}}^{\prime }}$is said to satisfy the condition (H1) (with respect toH0and$H_{0}^{\prime }$), if$\rho \circ ad{{H}_{0}}=adH_{0}^{\prime }\circ \rho$, and the condition (H2), if $\rho ({{H}_{0}})=H_{0}^{\prime }(\text{II,}\S \text{8)}$. For brevity, such a homomorphism is also called an (Hi)-homomorphism (i=1, 2). In this chapter, we will study (non-trivial) (H1)-homomorphisms$\kappa :\mathfrak{s}{{\mathfrak{l}}_{2}}(R)\to \mathfrak{g}$in their connection with the “boundary components” of the symmetric domain$\mathfrak{D}$associated with$\mathfrak{g}$. We shall see that such a homomorphism gives rise to a Siegel domain expression of$\mathfrak{D}$relative to the...

9. Chapter IV Equivariant Holomorphic Maps of a Symmetric Domain into a Siegel Space
(pp. 165-208)

LetGbe a group. A (linear) representation ofGoverFis a pair (V,ρ) formed of a finite-dimensional vector spaceVoverFand a group homomorphismρ:GGL(V). Sometimesρalone is called a representation andVis referred to as a representation-space. When we are dealing with anF-groupG, it will tacitly be assumed thatρis anF-homomorphism ofF-group (called sometimes an “F-representation”).

Irreducible representations, direct sums, etc. are defined in the usual manner. A representation (V,ρ) ofGis calledfully reducibleif it is a direct sum of...

10. Chapter V Infinitesimal Automorphisms of Symmetric Siegel Domains
(pp. 209-266)

Letֆ=ֆ(U,V,Ω,H) be a Siegel domain (of the second kind) defined by the following data:

$\left\{ {\begin{array}{*{20}{l}} {U = {\text{a real vector space of dimension }}m,}\\ {V = {\text{a complex vector space of dimension }}n,}\\ {\Omega = {\text{a (non - degenerate) open convex cone in }}U{\text{ (cf}}{\text{. I, \S 8),}}}\\ {H = {\text{a }}\Omega {\text{ - positive" hermitian map, i}}{\text{. e}}{\text{., a sesquilinear map }}V \times V \to {U_c}{\text{ satisfying the condition}}} \end{array}} \right.$$v\epsilon V,v\ne 0\Rightarrow H(v,v)\epsilon \bar{\Omega }-\text{ }\!\!\{\!\!\text{ 0 }\!\!\}\!\!\text{ }.$

(As before, we assume thatHisC-linear in the second variable andC-antilinear in the first.) Then one has

(1.1) $s=\{(u,v)\epsilon {{U}_{c}}\times V|\operatorname{Im}u-H(v,v)\epsilon \Omega \}.$

WhenVreduces to {0},ֆ=U+is a “tube domain” (or a Siegel domain of the first kind). We also use the following notation introduced previously:

Hol(ֆ) = the Lie group of holomorphic automorphisms ofֆ,

Aff(ֆ) = the Lie group of affine automorphisms ofֆ,

G(Ω) = the Lie group of linear automorphisms ofΩ,

$\mathfrak{g}=\mathfrak{g}(s)=\text{Lie Hol(s}),\mathfrak{g}(\Omega )=\text{Lie }G(\Omega )$

In this chapter,...

11. Appendix: Classical Domains
(pp. 267-290)
12. References
(pp. 291-318)
13. Index
(pp. 319-322)
14. Back Matter
(pp. 323-323)