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A remark on differential operator algebras and an equivalence of categories. (English) Zbl 0842.17009
Let \(G\) be a complex connected reductive algebraic group and \(P\) a parabolic subgroup of \(G\) containing a maximal torus \(H\). Denote their Lie algebras by \({\mathfrak g}\), \({\mathfrak p}\), \({\mathfrak h}\). To \(\lambda\in ({\mathfrak p}^*)^P\) one can associate, in a standard way, a sheaf of \(G\)-equivariant differential operators on \(G/P\) with ring of global sections \(A_\lambda\). There is a natural map \(\Phi_\lambda: U({\mathfrak g})\to A_\lambda\). One is interested in when this map is surjective; it need not be.
To explain the author’s result we need some notation. One regards \(\lambda\) as an element of \({\mathfrak h}^*\). Now let \(\Delta\) denote the set of positive roots of \({\mathfrak h}\) in the Lie algebra of a Levi factor of the commutator subgroup of \(P\) and let \(\rho\) denote the half-sum of the elements of \(\Delta\). It is shown that if \(\lambda\) is dominant integral on \(\Delta\) and \(\lambda+ \rho\) is dominant then \(\Phi_\lambda\) is surjective. This proves a result announced by Vogan. According to the author, Vogan has not published a proof of his assertion. One can identify \(A_\lambda\) with the module of \({\mathfrak g}\)-finite endomorphisms of a certain generalized Verma module. Usually, one imposes a \({\mathfrak p}\)-antidominance condition on \(\lambda+ \rho\) (rather than dominance) so that the generalized Verma module is simple which gives a surjectivity. So the point is that the author has managed to alter this hypothesis.

MSC:
17B35 Universal enveloping (super)algebras
16S30 Universal enveloping algebras of Lie algebras
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References:
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