Invariant differential operators are linear combinations of symmetric positive ones. (English) Zbl 0810.22006

Let \(E\) be a line bundle over the noncompact Hermitian symmetric space \(G/K\), and \(D_ G(E)\) the algebra of \(G\)-equivariant differential operators on \(E\). This paper proves that the algebra \(D_ G(E)\) has a basis consisting of symmetric and positive elements. More precisely, to each constituent \(Z\) of \(S({\mathfrak p})\), where \(\mathfrak p\) is the complexification of the tangent space of \(G/K\) at \(eK\) and \(S({\mathfrak p})\) denotes its symmetric algebra, a symmetric and positive operator \(D_ Z\) in \(D_ G(E)\) can be attached in a canonical way, and under this correspondence there are \(Z_ 1, \dots, Z_ r\) \((r = \text{rank } G/K)\) such that \(D_{Z_ 1}, \dots, D_{Z_ r}\) are algebraically independent and generate \(D_ G(E)\) as \(\mathbb{C}\)-algebra. For the special case of classical Hermitian symmetric spaces, G. Shimura poved the same result, but his algebraic approach is different from that of this paper.


22E46 Semisimple Lie groups and their representations
43A85 Harmonic analysis on homogeneous spaces
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
Full Text: DOI EuDML


[1] Deitmar, A.: Invariant operators on higherK-types, J. Reine Angew. Math412, 97-107 (1990) · Zbl 0712.43006
[2] Helgason, S.: Groups and geometric analysis, London New York: Academic Press 1984 · Zbl 0543.58001
[3] Schmid, W.: Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen R?umen, Invent. Math.9, 62-80 (1969) · Zbl 0219.32013
[4] Shimura, G.: Invariant differential operators on hermitian symmetric spaces, Ann. Math.132, 237-272 (1991) · Zbl 0718.11020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.