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In this paper, we extend recent work of one of us [Br] to investigate an old problem of the other one [B2]. Given a connected semisimple complex Lie-group G with Lie-algebra g, we study the representation $$\psi _X :U(\mathfrak{g}) \to D(X)$$ of the enveloping algebra of $$\mathfrak{g}$$ by global differential operators on a complete homogeneous space X=G/P. It turns out that the kernel I of ψ is the annihilator of a generalized Verma-module. On the other hand, we study the associated graded ideal gr I, and relate it to the geometry of a generalized Springer-resolution, that is a map $$\pi _X :T^* (X) \to \mathfrak{g}$$ of the cotangent-bundle of X onto a nilpotent variety in $$\mathfrak{g}$$ , as studied e.g. in [BM1]. We prove, for instance, that gr I is prime if and only if π is birational with normal image. In general, we show that $$\sqrt {grI_X }$$ is prime. Equivalently, the associated variety of I in $$\mathfrak{g}$$ is irreducible: In fact, it is the closure of the Richardson-orbit determined by P. For the homogeneous space Y=G/(P, P), we prove that the analogous ideal I has for associated variety the closure of the Dixmier-sheet determined by P. From this main result, we derive as a corollary, that for any module induced from a finitedimensional Lie P-module the associated variety of the annihilator is irreducible, proving an old conjecture [B2], 2.5. Finally, we give some applications to the study of associated varieties of primitive ideals. [ABSTRACT FROM AUTHOR]

Let g be a complex Lie-algebra, G its adjoint algebraic group, and the space of primitive ideals in the enveloping algebra of ℷ. For solvable ℷ, J. Dixmier has defined a map from the dual space g* into, and R. Rentschler has proved the injectivity of the corresponding map g*/ G→ defined on the space of orbits of G in g*. The aim of the present paper is to define a similar map g*/ G→ for g= sl( n, C). In a joint paper with J. C. Jantzen we shall prove that it is also injective. [ABSTRACT FROM AUTHOR]

Let g be a complex semi-simple Lie-algebra of finite dimension, U(g) its enveloping algebra with center Z(g), and the space of primitive ideals of U(g). Then J→ Jν Z(g) is a projection (with finite fibres) π: → Max Z(g) of onto the space of maximal ideals of Z(g). Let $$\mathfrak{h} \subset \mathfrak{g}$$ be a Cartan subalgebra, R the root-system, B a base, and for each γ∈ $$\mathfrak{h}^* $$ let L(λ) denote the simple g-module of highest weight λ. (By recent work of Duflo, λ→Ann L(λ) maps $$\mathfrak{h}^* $$ surjectively onto.) Let denote the π-fibre of containing the primitive ideal Ann L(λ). Our first main subject is to relate to for suitable integral weights v. This is done by a 'translation principle' (2.12), which is too complicated to be stated here in full detail, but which is illustrated by the following special case: If λ is inside some Weyl-chamber and λ+ v is in the same chamber, then there exists an isomorphism of ordered sets. Moreover, if the structure of is known for all λ inside some chamber, then it will be known in general. -This translation principle turns out to be an extremely useful tool for the study of. Some of the applications are: 1. a lower estimate for the cardinality of (2.15), 2. a description of the 'almost maximal' elements of (2.19), 3. the existence of many 'non-induced' maximal ideals similar to Joseph's one (4.2), 4. the existence of another type of 'non-induced' ideals which occurs already for $$\mathfrak{g} = \mathfrak{s}\mathfrak{l}_n $$ , n=4, 5,...(4.4). Our second main subject is the study of Dixmier's orbit-map [8, 9] which relates orbits of the adjoint group in the dual space of g to primitive ideals. We prove that this map is injective for $$\mathfrak{g} = \mathfrak{s}\mathfrak{l}_n $$ (5.15). For this purpose, we study infinite intersections of maximal and of 'induced' ideals of U(g) and combine these techniques with our translation principle. [ABSTRACT FROM AUTHOR]