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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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