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Invariant differential operators on Hermitian symmetric spaces. (English) Zbl 0718.11020

Let G be a connected noncompact semisimple Lie group with finite center and K a maximal compact subgroup of G. The author studies the ring of invariant differential operators on the Hermitian symmetric space G/K. More precisely he considers the ring \({\mathcal D}(\rho)\) of left-invariant differential operators attached to a one-dimensional representation \(\rho\). The complexification \({\mathfrak g}\) of the Lie algebra of G has abelian subalgebras \(\wp_+\) and \(\wp_ -\). Each irreducible constituent Z of the space of complex valued homogeneous polynomial functions of degree r on \(\wp_+\) gives rise to an element \({\mathcal L}_ Z\in {\mathcal D}(\rho)\). Moreover \((-1)^ r {\mathcal L}_ Z\) is symmetric and nonnegative with respect to a Hermitian inner product. There exists a canonically defined set of generators of the polynomial ring \({\mathcal D}(\rho)\). Moreover the nonnegativity leads to various inequalities for the eigenvalues of the generators on a common eigenfunction.
Reviewer: A.Krieg (Münster)

MSC:

11F60 Hecke-Petersson operators, differential operators (several variables)
11F70 Representation-theoretic methods; automorphic representations over local and global fields
32N15 Automorphic functions in symmetric domains
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