# zbMATH — the first resource for mathematics

Continuation of unitary derived functor modules out of the canonical chamber. (English) Zbl 0582.22013
The authors present a method for proving unitarizability for representations of a real semi-simple group G obtained from certain $${\mathfrak g}$$-modules of its Lie algebra $${\mathfrak g}$$. The $${\mathfrak g}$$- modules considered are the Zuckerman derived functor modules obtained from generalized Verma modules and the method is that of an analytic continuation by means of varying the character of the Cartan subalgebra. This method relies on an earlier joint work of the authors with R. Parthasarathy and N. R. Wallach [Proc. Natl. Acad. Sci. USA 80, 7047-7050 (1983; Zbl 0527.22007)] where a general criterion for obtaining unitary representations that way was proved.
Here an account of a computational procedure is presented enabling an application of this general result to classical simple groups and, by means of appended computer programs, to exceptional groups $$E_ 6$$, $$E_ 7$$, $$E_ 8$$. The new unitary representations obtained are tabulated in the appendix. Use is made of Jantzen’s irreducibility criterion [J. C. Jantzen, Math. Ann. 226, 53-65 (1977; Zbl 0372.17003)].
Reviewer: A.Strasburger

##### MSC:
 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 22E46 Semisimple Lie groups and their representations 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 22-04 Software, source code, etc. for problems pertaining to topological groups
Full Text:
##### References:
 [1] A. Borel and J. de Siebenthal , “ Les sous-groupes fermés de rang maximum des groupes de Lie clos ,” Comment. Math. Helv. 23 ( 1949 ), 200-221. MR 11,326d | Zbl 0034.30701 · Zbl 0034.30701 [2] T.J. Enright , “ Unitary representations for two real forms of a semisimple Lie algebra : a theory of comparison ,” Proceedings of the College Park Conference, 1982 , to appear. Zbl 0531.22012 · Zbl 0531.22012 [3] T.J. Enright , R. Howe and N.R. Wallach , “ A classification of unitary highest weight modules ,” Proceedings of the Park City Conference on Representations of Reductive Groups, 1982 , to appear. Zbl 0535.22012 · Zbl 0535.22012 [4] T.J. Enright , R. Parthasarathy , N.R. Wallach and J.A. Wolf , “ Classes of unitarizable derived functor modules ,” Proc. Nat. Acad. Sci. USA, 1983 , to appear. Zbl 0527.22007 · Zbl 0527.22007 [5] T.J. Enright , R. Parthasarathy , N.R. Wallach and J.A. Wolf , “ Unitary derived functor modules with small spectrum ,” to appear. Zbl 0568.22007 · Zbl 0568.22007 [6] J.C. Jantzen , “ Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren ,” Math. Ann. 226 ( 1977 ), 53-65. MR 55 #12783 | Zbl 0372.17003 · Zbl 0372.17003 [7] N.R. Wallach , “ The analytic continuation of the discrete series, I, II ,” Trans. Amer. Math. Soc. 251 ( 1979 ), 1-17 and 19-37. MR 81a:22009 | Zbl 0419.22017 · Zbl 0419.22017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.