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Modules of \({\mathfrak k}\)-finite vectors over semi-simple Lie algebras. (English) Zbl 0543.17004

Let \({\mathfrak g}\) be a complex semisimple Lie algebra. If M and N are modules for \({\mathfrak g}\) (and hence for the enveloping algebra \(U({\mathfrak g}))\), then \(Hom_{{\mathbb{C}}}(M,N)\) is a \({\mathfrak g}\times {\mathfrak g}\) module in an obvious way. Write \({\mathfrak k}\) for the diagonal in \({\mathfrak g}\times {\mathfrak g}\), and L(M,N) for the subspace of k-finite vectors in the Hom. Then L(M,M) is an algebra, and L(M,N) is an \((L(M,M),L(N,N))- bimodule.\) The action of \(U({\mathfrak g})\) on M defines a homomorphism from \(U({\mathfrak g})\) into L(M,M).
The purpose of this paper is to study L(M,N), with emphasis on the case when M and N have finite length as \({\mathfrak g}\) modules. The goal is to show that its structure is similar or identical to that found in the special case when M and N are highest weight modules. (The fundamental result of this type is Duflo’s theorem that if M is simple, then there is a simple highest weight module X such that M and X have the same annihilator in \(U({\mathfrak g}).)\)
A typical result of this paper is that if M is simple, then L(M,M) is a maximal order in its ring of fractions. Since rings of fractions are fairly rigid objects, this leads to comparison theorems of the kind sought; they are not completely general, however, and they are a little complicated to state here. A minor objective also discussed by the authors is to show that primitive quotients of \(U({\mathfrak g})\) are themselves maximal orders; but this is now known to be false in general.
The paper also contains other ring-theoretic results about primitive quotients of \(U({\mathfrak g})\), such as a construction of many projective modules for them (and hence a discussion of the Grothendieck groups \(K_ 0)\).
Reviewer: D.Vogan

MSC:

17B35 Universal enveloping (super)algebras
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