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Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras. (English) Zbl 0445.17006
Let $$\mathfrak g$$ be a complex semisimple Lie algebra, $$U(\mathfrak g)$$ its enveloping algebra, $$Z$$ the center of $$U(\mathfrak g)$$. Let $$V$$ be the finite-dimensional $$\mathfrak g$$-module.
The following was known and was used for a long time. If $$M$$ is any $$\mathfrak g$$-module such that $$Z$$ acts on $$M$$ by the scalars via the character $$\theta$$ of $$Z$$, and if $$\theta'$$ is any other character of $$Z$$, then some properties of the $$\theta'$$-component of $$F_V(M)\cong V\otimes M$$ can be decuced from those of $$M$$. The useful trick (this is often the method in future and the aim of this paper) is to describe this method in the possible generality.
The first and most interesting question about the functor $$F_V$$ is the decomposition as the straight sum of the indecomposable ones. (The straight summands of $$F_V$$ will be called the “projective” functors.) Being considered on the category $$\mathcal M_{Zf}$$ of $$Z$$-finite $$\mathfrak g$$-modules, $$F_V$$ has the well-known summands $$\operatorname{Pr}(\theta) \circ F_V \circ \operatorname{Pr}(\theta')$$ but they are often decomposable, too. (Here $$\operatorname{Pr}(\theta)M$$ is the $$\vartheta$$-component of $$M)$$.
The main result of Chapter 1 of the paper is the description of all the projective functors and their morphisms. This classification is based on the theorem below. Let $$\mathfrak h$$ be the Cartan subalgebra of $$\mathfrak g$$, $$\rho$$ the half-sum of the positive roots of the pair $$(\mathfrak g,\mathfrak h)$$ for some ordering and $$n$$ the sum of the root subspaces of $$\mathfrak g$$ for the positive roots. Denote by $$M_\chi$$ the Verma module $$U(\mathfrak g)/U(\mathfrak g)(n +I_{\chi+\rho})$$ with the highest weight $$\chi\in \mathfrak h^*$$. Here $$I_{\chi+\rho}$$ is the ideal of $$U(\mathfrak h)$$, generated by the elements $$h-(\chi-\rho)(h)$$, $$h\in\mathfrak h$$. Let $$\eta\colon \mathfrak h^*\to \hat Z$$ be the Harish-Chandra mapping of $$\mathfrak h^*$$ to the set of all characters of $$Z$$. Let $$\mathcal M(\vartheta)$$ be the subcategory of $$\mathcal M_{Zf}$$ of $$\mathfrak g$$-modules with eigenvalues $$\theta(z)$$, $$z\in Z$$ for the character $$\theta$$ of $$Z$$.
Theorem 3.5. Let $$F$$, $$G$$ be the projective $$\theta$$-functors (the straight summands of the functors $$F_V\vert_{\mathcal M(\theta)})$$. $$\chi\in \eta^{-1}(\theta)$$ be the weight. Then the canonical homomorphism
$i_\chi\colon \operatorname{Hom}(F,G) \rightarrow \operatorname{Hom}(FM_\chi, GM_\chi)$ is a monomorphism and an isomorphism if $$\chi$$ is dominant.
The authors deduce from this that the restriction of the projective functor to $$\mathcal M(\theta)$$ is defined by the $$\mathfrak g$$-module $$F(M_\chi)$$. For example, the straight summands of $$F$$ correspond to those of $$F(M_\chi)$$. The task of the decomposition of the module $$F(M_\chi)$$ as the sum of indecomposable$$\mathfrak g$$-modules is more simple.
Then the authors use the description of indecomposable projective functors in order
1) to prove that the functor $$\operatorname{Pr}(\theta) \circ F_V \circ \operatorname{Pr}(\theta')$$ establishes the equivalence between the categories $$\mathcal M(\theta)$$, $$\mathcal M(\theta')$$ for certain pairs $$(\theta,\theta')$$ (this refines the corresponding results of G. Zuckermann, T. Enright and D. Vogan);
2) to relate the two-sided ideals of $$U(\mathfrak g)$$ and the submodule lattice of Verma modules (this reproduces the results of A. Joseph) and
3) to study the multiplicities in Jordan-Hölder series of Verma modules.
In Chapter 2 the authors use the classification of projective functors to study the Harish-Chandra modules over the complex semisimple Lie group with Lie algebra $$\mathfrak g$$. The possibility of this application is based on the fact, that each Harish-Chandra module $$M$$, considered as the $$(U(\mathfrak g),U(\mathfrak g))$$-bimodule, is the factor module of the bimodule $$V \otimes U(\mathfrak g)$$, where $$V$$ is the finite-dimensional $$\mathfrak g$$-module. So, the functor $$Y \rightarrow M\otimes Y$$ in the category $$\mathcal M$$ is the functor of the functor $$F_V$$.
The simple and purely algebraic way, presented here, leads to the classification of irreducible Harish-Chandra modules over the group $$G_{\mathbb C}$$ with Lie algebra $$\mathfrak g$$ (first obtained by D. Zhelobenko by analytical methods), and to the equivalence between the category of Harish-Chandra modules over $$G_{\mathbb C}$$ with given $$Z$$-character and some subcategory of the category $$\mathcal O$$ of $$\mathfrak h$$-semisimple, $$U(n)$$-finite $$\mathfrak g$$-modules. This equivalence transforms the principal series modules to the dual of the Verma modules. Thus, the authors redefine the coincidence of the multiplicities of the Jordan-Hölder series of these types of $$\mathfrak g$$-modules (obtained earlier by M. Duflo).
Reviewer: S. Prishchepionok

##### MSC:
 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B20 Simple, semisimple, reductive (super)algebras 22E46 Semisimple Lie groups and their representations 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 17B55 Homological methods in Lie (super)algebras
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