## 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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