## 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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