Contemporary Geometry pp 169-185 | Cite as

# Dimensions of the Rings of Invariant Differential Operators on Bounded Homogeneous Domains

Chapter

## Abstract

Let
where , \(z=\left( {{z}_{1}},\ldots ,{{z}_{r}} \right),\omega =\left( {{\omega }_{1}}\ldots ,{{\omega }_{n}} \right),\),and

*D*be a bounded homogeneous domain in ℂ^{ n }, and suppose*G(D)*is the identity component of the analytic automorphism group of*D*. By a famous theorem of H. Cartan,*G(D)*is a real Lie group. We might as well assume that 0 ∈*D*. From a well-known result in function theory of several complex variables the isotropy group*G*_{ o }*(D)*of*G(D)*at 0 ∈*D*has the following representation:$$\omega =zA+o(z),$$

(6.1)

*A*is an*n*×*n*complex matrix constituting the linear parts {*A*}of*G*_{ o }*(D)*, which form a compact subgroup of the unitary group*U(n)*denoted by*K.*## Keywords

Irreducible Representation Compact Subgroup Homogeneous Part Classical Domain Irreducible Decomposition
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

- 1.L. K. Hua,
*Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains*, American Mathematical Society, Providence,**RI**(1963).Google Scholar - 2.Q. K. Lu (K. H. Look),
*Introduction to Function Theory of Several Complex Variables*, China Academic, Beijing (1961) (in Chinese)Google Scholar - 3.Q. K. Lu,
*Classical Manifolds and Classical Domains*, Shanghai Science and Technical publishers, China (1963) (in Chinese)Google Scholar - 4.Shingo Murakami, On automorphisms of Siegel domains, in Lecture Notes in Math., vol. 286, Springer-Verlag, New York (1972).Google Scholar
- 5.A. Selberg, Harmonic analysis and discontinuous group in weakly symmetric Riemannian space with applications to Dirichlet series,
*J. Ind. Math. Soc*.**20**(1965).Google Scholar

## Copyright information

© Plenum Press, New York 1991