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Zbl 0842.17009
McGovern, William M.
A remark on differential operator algebras and an equivalence of categories. (English)
[J] Compos. Math. 90, No.3, 305-313 (1994). ISSN 0010-437X; ISSN 1570-5846/e

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 ${\germ g}$, ${\germ p}$, ${\germ h}$. To $\lambda\in ({\germ p}^*)^P$ 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 $\Phi_\lambda: U({\germ g})\to A_\lambda$. One is interested in when this map is surjective; it need not be. \par To explain the author's result we need some notation. One regards $\lambda$ as an element of ${\germ h}^*$. Now let $\Delta$ denote the set of positive roots of ${\germ h}$ 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 ${\germ g}$-finite endomorphisms of a certain generalized Verma module. Usually, one imposes a ${\germ p}$-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.
[M.P.Holland (Sheffield)]

MSC 2000:: *17B35 Universal enveloping algebras (Lie algebras)
16S30 Universal enveloping algebras of Lie algebras (associative)

Keywords: equivariant twisted differential operators; parabolic subgroups; generalized flag varieties; surjectivity; finite vectors

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