Invariant differential operators on symmetric spaces. II: The generalized Harish- Chandra homomorphism
Let \(G/K\) be a symmetric space with symmetry \(\sigma\), and let \({\mathfrak g}= {\mathfrak k}+ {\mathfrak p}\) be the corresponding decomposition of the Lie algebra \({\mathfrak g}\) of \(G\). The paper studies the algebra \({\mathcal A}\) of invariant differential operators on the line bundle of half-densities on \(G/K\). The symbols of these operators make up the algebra \(S({\mathfrak p})^ K\) of \(K\)-invariant polynomials on the dual space of \({\mathfrak p}\). The algebra \({\mathcal A}\) can also be viewed as a quotient of the centralizer of \({\mathfrak k}\) in the universal enveloping algebra of \({\mathfrak g}\), and the classical symmetrization map gives a vector space isomorphism of \(S({\mathfrak p})^ K\) onto \({\mathcal A}\). But \({\mathcal A}\) is abelian, thus it is natural to look for an algebra isomorphism between \(S({\mathfrak p})^ K\) and \({\mathcal A}\). Such an isomorphism was known to exist in several important cases: for semisimple \(G\) (Harish-Chandra's isomorphism), for nilpotent \({\mathfrak g}\) (Benoist), or when \(G/K\) is a Lie group (Duflo's isomorphism). The author's paper makes an important step towards the general case: assuming \(G\) is a linear algebraic group with rational involution \(\sigma\), he constructs a generalized Harish-Chandra isomorphism of \({\mathcal A}\) onto Frac \(S({\mathfrak p})^ K\), the field of \(K\)-invariant fractions. It is conjectured that this is actually an isomorphism of \({\mathcal A}\) onto \(S({\mathfrak p})^ K\) itself. The proof combines several tools introduced in the first part of the paper [cf. the preceding review Zbl 0803.43003]. A major step is a reduction of the problem to the case when \(G/K\) is ``of nilpotent type'', to which Benoist's method can be extended.
scientific article; zbMATH DE number 515227