On the unitarizability of derived functor modules. (English) Zbl 0547.22008
Let G be a real reductive Lie group. Most of the unitary representations of G arise from unitary representations of smaller groups in one of two ways: parabolic induction, and cohomological parabolic construction. (The latter method is known as ”the” derived functor construction, which may lead to legitimate questions about where group representers have been for the last forty years.) Parabolic induction has been fairly well understood since the 1950’s; it produces things like principal series representations. Cohomological parabolic induction (which is not induction in Mackey’s sense at all) has its roots in the same period, in the Borel-Weil theorem. However, a complete theory for non-compact groups appeared only recently, in the work of Zuckerman. It produces things like the discrete series. (Of course the discrete series themselves had previously been found by Harish-Chandra, by very deep ad hoc methods.)
Zuckerman’s idea was to abandon the trappings of analysis entirely. Roughly speaking, he constructed not group actions on functions, but only Lie algebra actions on Taylor series. Using old results of Harish- Chandra, he deduced the existence of Hilbert space models for his representations; but it was not possible to make the group action unitary without more effort. T. J. Enright and the author soon succeeded in finding invariant Hermitian forms [Duke Math. J. 47, 1-15 (1980; Zbl 0429.17012)].
The problem which remained was to prove that these forms were positive. This was proved by the reviewer [Ann. Math. 120, 141-187 (1984)] in a simple but unsatisfactory way. (Essentially, the problem was reduced to the special case of the discrete series, where Harish-Chandra had provided the answer by indirect methods.)
In these few pages, the author gives a direct and self-contained proof. The argument is so simple that it is hard to pinpoint a main idea. After a little algebraic hand-waving (using, for example, the Euler-Poincaré principle), one is reduced to calculating the signature of a form in a Verma-type module. That problem in turn is deformed (using continuity of signature) to a trivial one.
This is a brilliant piece of work, and should be required reading for unitary group representers.
Zuckerman’s idea was to abandon the trappings of analysis entirely. Roughly speaking, he constructed not group actions on functions, but only Lie algebra actions on Taylor series. Using old results of Harish- Chandra, he deduced the existence of Hilbert space models for his representations; but it was not possible to make the group action unitary without more effort. T. J. Enright and the author soon succeeded in finding invariant Hermitian forms [Duke Math. J. 47, 1-15 (1980; Zbl 0429.17012)].
The problem which remained was to prove that these forms were positive. This was proved by the reviewer [Ann. Math. 120, 141-187 (1984)] in a simple but unsatisfactory way. (Essentially, the problem was reduced to the special case of the discrete series, where Harish-Chandra had provided the answer by indirect methods.)
In these few pages, the author gives a direct and self-contained proof. The argument is so simple that it is hard to pinpoint a main idea. After a little algebraic hand-waving (using, for example, the Euler-Poincaré principle), one is reduced to calculating the signature of a form in a Verma-type module. That problem in turn is deformed (using continuity of signature) to a trivial one.
This is a brilliant piece of work, and should be required reading for unitary group representers.
Reviewer: D.Vogan
MSC:
| 22E47 | Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 17B55 | Homological methods in Lie (super)algebras |
Keywords:
positive form; real reductive Lie group; unitary representations; discrete series; Euler-Poincaré principle; signatureCitations:
Zbl 0429.17012References:
| [1] | Borel, A., Wallach, N.: Continuous cohomology discrete subgroups, and representations of reductive groups. Annals of Math. Study 94, Princeton: University Press 1980 · Zbl 0443.22010 |
| [2] | Enright, T., Parthasarthy, R., Wallach, N., Wolf, J.: Unitary derived functor modules with small spectrum. To appear Acta Math. · Zbl 0568.22007 |
| [3] | Enright, T., Wallach, N.: Notes on homological algebra and representations of Lie algebras. Duke Math. J. 47, 1-15 (1980) · Zbl 0429.17012 · doi:10.1215/S0012-7094-80-04701-8 |
| [4] | Rocha, A., Wallach, N.: Characters of irreducible representations of the Lie algebra of vector fields on the circle. Invent. Math. 72, 57-75 (1983) · Zbl 0498.17010 · doi:10.1007/BF01389129 |
| [5] | Vogan, D.: Representations of real reductive Lie groups., Progress in Mathematics, Vol. 15. Boston: Birkhäusen 1981 · Zbl 0469.22012 |
| [6] | Vogan, D. Understanding the unitary dual, Proceedings, University of Maryland 1982-1983. Lecture notes in Mathematics, Vol. 1024, 264-286. Berlin-Heidelberg-New York: Springer 1983 |
| [7] | Vogan, D.: Unitarizability of certain series of representations. To appear Ann. Math. · Zbl 0561.22010 |
| [8] | Vogan, D., Zuckerman, G.: Unitary representations with non-zero cohomology. To appear. · Zbl 0692.22008 |
| [9] | Wallach, N.R.: Real reductive groups. Boston: Birkhäuser (To appear) · Zbl 0666.22002 |
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