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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.
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
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