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Some results on invariant differential operators on symmetric spaces. (English) Zbl 0838.43012

Let \(G\) be a connected, noncompact semisimple Lie group with finite center, \(K\) a maximal compact subgroup and \(X= G/K\) the associated symmetric space. Let \({\mathbf D}(G/K)\) denote the algebra of differential operators on \(X\) which are invariant under the transformations \(\tau(g): hK\to ghK\) of \(X\), \(g\in G\) being arbitrary. We investigate connections between the algebra \({\mathbf D}(G/K)\) and the algebra \({\mathbf Z}(G)\) of differential operators on \(G\) which are invariant under all left and right translations.
To state the results let \({\mathbf D}(G)\) denote the algebra of all left-invariant differential operators on \(G\), \({\mathbf D}_K(G)\) the subalgebra of \({\mathbf D}(G)\) consisting of those operators which are also invariant under right translations from \(K\). Let \(\mu: {\mathbf D}_K(G)\to {\mathbf D}(G/K)\) denote the homomorphism given by \((\mu(D)f)\circ \pi= D(f\circ \pi)\), where \(\pi: G\to G/K\) is the natural map and \(f\in C^\infty(X)\). Since \(G/K\) is reductive, \(\mu\) is surjective [the author, Acta Math. 102, 239-299 (1959; Zbl 0146.43601)].
We consider another homomorphism \(\nu\) of \({\mathbf D}(G)\) into the algebra \({\mathbf E}(X)\) of all differential operators on \(X\). The map which to \(g\in G\) assigns the linear transformation \(f\to f\circ \tau(g^{- 1})\) of \(C^\infty(X)\) is a representation of \(G\) whose differential is the indicated homomorphism \(\nu: {\mathbf D}(G)\to {\mathbf E}(X)\). It is known [the author, Am. J. Math. 86, 565-601 (1964; Zbl 0178.17001)] that \(\nu(D)= \mu(D^*)\), \(D\in {\mathbf Z}(G)\), the asterisk denoting adjoint. The image of \({\mathbf Z}(G)\) under \(\mu\) (and \(\nu\)) will be denoted by \({\mathbf Z}(G/K)\). We shall prove that each \(D\in {\mathbf D}(G/K)\) is a “quotient” of two members \(Z_1, Z_2\in {\mathbf Z}(G/K)\), that is \(DZ_1= Z_2\), \((Z_1\neq 0)\).
For \(G\) classical (real or complex) we have actually \({\mathbf D}(G/K)= {\mathbf Z}(G/K)\) [the author, 1964, op. cit.] and we shall see in the present paper that this remains true for all noncompact irreducible symmetric spaces with exactly four exceptions, namely the symmetric pairs \[ ({\mathfrak e}_6, {\mathfrak s}{\mathfrak o}(10)+ {\mathbf R}),\;({\mathfrak e}_6, {\mathfrak f}_4),\;({\mathfrak e}_7, {\mathfrak e}_6+ {\mathbf R}),\;({\mathfrak e}_8, {\mathfrak e}_7+ {\mathfrak s}{\mathfrak u}(2)). \] The proofs are based on relating these differential operators to Weyl group invariants.

MSC:

43A85 Harmonic analysis on homogeneous spaces
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
53C35 Differential geometry of symmetric spaces
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