Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras. (English) Zbl 0445.17006

Bernstein, J. N.; Gelfand, S. I.

Compos. Math. 41, 245-285 (1980).

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

Keywords:

infinite dimensional representations; semisimple Lie algebra; Verma modules; tensor product; projective functor; irreducible Harish-Chandra modules; principal series modules; category of finite modules; category O

References:

[1]	C. Zuckerman : Tensor products of finite and infinite dimensional representations of semisimple Lie groups . Ann. of Math. 106 (1977) 295-308. · Zbl 0384.22004
[2]	J.N. Bernstein , I.M. Gelfand and S.I. Gelfand : The structure of representations generated by vectors of highest weight , Funk. Anal. Appl. 51 (1971), N1, 1-9 (Russian). · Zbl 0246.17008
[3]	W. Borho and J.C. Jantzen : Uber primitive Ideale in der Einhullenden einer halbeinfachen Lie-Algebra . Inventiones Math. 39 (1977) 1-53. · Zbl 0327.17002
[4]	N.R. Wallach : On the Enright-Varadarajan modules , Ann. Sci. Ecole Norm. Sup. 9 (1976) 81-101. · Zbl 0379.22008
[5]	A.W. Knapp and N.R. Wallach : Szegö kernels associated with discrete series , Inventiones Math. 34 (1976) 163-200. · Zbl 0332.22015
[6]	W. Schmidt : L2-cohomology and the discrete series , Ann. of Math. 103 (1976) 375-394. · Zbl 0333.22009
[7]	M.F. Atiyah and W. Schmidt : A geometric construction of the discrete series for semisimple Lie groups , Inventiones Math. 42 (1977) 1-62. · Zbl 0373.22001
[8]	B. Kostant : On tensor product of a finite and infinite dimensional representation . Jour. Func. Anal. v. 20 N 4 (1975) 257-285. · Zbl 0355.17010
[9]	M. Duflo : Sur la classification des ideaux primitifs dans l’algèbre envelloppante d’une algèbre de Lie semi-simple . Ann. of Math. 105 (1977) 107-120. · Zbl 0346.17011
[10]	D.P. Zhelobenko : Harmonic analysis on semisimple complex Lie groups , Moscow, ”Nauka”, 1974 (Russian). · Zbl 0204.14403
[11]	M. Duflo : Représentations irréductibles des groupes semisimples complexes , Lecture Notes in Math. 497 (1975) 26-88. · Zbl 0315.22008
[12]	M. Duflo and N. Conze-Berline : Sur les représentations induites des groupes semi-simple complexes , Compositio Math. 34 (1977) 307-336. · Zbl 0389.22016
[13]	H. Bass : Algebraic K-theory , Benjamin, 1968. · Zbl 0174.30302
[14]	B. Mitchell : Theory of categories , Academic Press, N.Y., 1965. · Zbl 0136.00604
[15]	S. Maclane : Categories for the Working Mathematician , Springer, New York-Berlin-Heidelberg, 1971. · Zbl 0232.18001
[16]	J. Dixmier : Algèbres envelloppantes . Paris, Gauthier-Villars, 1974. · Zbl 0308.17007
[17]	N. Bourbaki : Groupes et algèbres de Lie , Ch. 4, 5, 6. Paris, Hermann, 1968. · Zbl 0483.22001
[18]	B. Kostant : Lie group representations on polynomial rings , Amer. J. Math. 81 (1959) 937-1032. · Zbl 0099.25603
[19]	J.N. Bernstein , I.M. Gelfand and S.I. Gelfand : On a category of G-modules . Funk. Anal. Appl. 10 (1976) N2, 1-8 (Russian). · Zbl 0353.18013
[20]	B. Kostant : A formula for the multiplicity of a weight , Trans. Amer. Math. Soc. 93 (1959) 53-73. · Zbl 0131.27201
[21]	J. Lepowsky and N.R. Wallach : Finite- and infinite-dimensional representations of linear semisimple groups , Trans. Amer. Math. Soc. 184 (1973) 223-246. · Zbl 0279.17001
[22]	D. Vogan : Irreducible Characters of Semisimple Lie Groups I , Duke Math. J. 46 (1979) 61-00. · Zbl 0398.22021
[23]	J.C. Jantzen : Moduln mit einem höchsten Gewicht , Habilitationsschrift, Universität Bonn, 1977. · Zbl 0426.17001
[24]	T. Enright : On the Irreducibility of the Fundamental Series of a Real semisimple Lie Algebra , Ann. of Math. 110 (1979) 1-82. · Zbl 0417.17005
[25]	T. Enright : The Representations of Complex Semisimple Lie Groups (preprint).
[26]	A. Joseph : Dixmier’s Problem for Verma and Principal Series Submodules (preprint), December, 1978. · Zbl 0421.17005