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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.
MSC:
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)
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References:
[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
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