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The unitary spectrum for real rank one groups. (English) Zbl 0561.22009
A complete description is given of the irreducible unitary representations of semisimple Lie groups of real rank one. Except for the case of the exceptional group \(F_{4,1}\) the representations were known before. However the advantage of the paper under review is that the problem is treated from a rather general point of view. Thus many of the procedures needed for the higher rank case are introduced and used instead of case by case calculations. Of cource the problem of determining the unitary spectrum for groups of higher rank being one of today’s outstanding problems one cannot expect the known general procedures to give the full result. But these procedures allow one to reduce the problem to either the determination of the unitary spherical representations of a subgroup, which is solved by B. Kostant [Lie Groups Represent., Budapest 1971, 231-329 (1975; Zbl 0327.22010)], or to the discussion of certain special representations of the groups Sp(n,1) and \(F_{4,1}.\)
Corollary 3.5 gives an affirmative answer for groups of real rank one to a conjecture of Zuckermann that a certain construction of irreducible representations via derived functors always leads to unitary representations. This conjecture has later been proved in general in D. Vogan [Ann. Math., II Ser. 120, 141-187 (1984; review below, Zbl 0561.22010)]. As another application the unitary representations that contribute to the \(L^ 2\)-index of the Dirac operator are determined.
The paper only treats linear groups with a compact Cartan subgroup with the excuse that the other cases are known already, and probably easier. Anyway I find this a minor weak point, since it would have been nice, when trying to persue general methods, to incorporate into the treatment all real rank one groups, linear or nonlinear, with or without a compact Cartan subgroup. The paper contains two many small misprints.
Reviewer: M.Flensted-Jensen

MSC:
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
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References:
[1] Atiyah, M.F., Schmid, W.: A geometric construction of the discrete series. Invent. Math.42, 1-62 (1977) · Zbl 0373.22001
[2] Baldoni Silva, M.W.: The unitary dual ofSp(n, 1), n?2. Duke Math. Journal48, 549-583 (1981) · Zbl 0496.22019
[3] Baldoni Silva, M.W., Kraljevic, H.: Composition factors of the principal series representations of the groupsSp(n, 1). TAMS262, 447-471 (1980) · Zbl 0448.22013
[4] Borel, A., Wallach, N.: Continuous cohomology, Discrete subgroups and representations of reductive groups. Ann. of Math. Studies, 94, Princeton University Press 1980 · Zbl 0443.22010
[5] Hirai, T.: On irreducible representations of the Lorentz group ofn-th order. Proc. Japan Acad.38, 83-87 (1962) · Zbl 0105.09703
[6] Johnson, K., Wallach, N.: Composition series and intertwining operators for the spherical principal series I. TAMS229, 137-173 (1977) · Zbl 0349.43010
[7] Kraljevic, H.: Representations of the universal covering group of the groupSU(n, 1). Glasnik Mat.8, 23-72 (1973) · Zbl 0262.22011
[8] Kostant, B.: On the existence and irreducibility of certain series of representations. Bull. A.M.S.75, 627-642 (1969); Lie groups and their representations (Summer School of the Bolyal János Mathematical Society), Halsted Press, New York 1975, pp. 231-329 · Zbl 0229.22026
[9] Knapp, A., Stein, E.M.: Intertwining operators for semisimple groups. Ann. of Math.93, 489-578 (1971) · Zbl 0257.22015
[10] Knapp, A., Zuckerman, G.: Classification theorems for representations of semisimple Lie groups. Lecture Notes in Mathematics, Vol. 587, pp. 138-159, Berlin-Heidelberg-New York: Springer · Zbl 0353.22011
[11] Langlands, R.: On the classification of irreducible representations of real algebraic groups. Mimeographed notes, Inst. for Advanced Study, 1973
[12] Moscovici, H.:L 2-index of elliptic operators on locally symmetric spaces of finite volume. Preprint · Zbl 0495.58026
[13] Miatello, R.: Alternating sum formulas for multiplicities inL 2(?/G) I and II. manuscript
[14] Speh, B., Vogan, D.: Reducibility of generalized principal series representations. Acta Math.145, 227-299 (1980) · Zbl 0457.22011
[15] Thieleker, E.: On the quasi-simple irreducible representations of the Lorentz groups. TAMS179, 465-505 (1973) · Zbl 0258.22016
[16] Vogan, D.: The algebraic structure of the representation of semisimple Lie groups I. Ann. of Math.109, 1-60 (1979) · Zbl 0424.22010
[17] Vogan, D.: Representations of real reductive Lie groups. Progress in Mathematics, Birkhauser, 1981 · Zbl 0469.22012
[18] Vogan, D.: Singular unitary representations. Lecture Notes in Math, vol. 880, pp. 506-535. Berlin-Heidelberg-New York: Springer 1980
[19] Vogan, D.: Unitary representations with cohomology. Preprint · Zbl 0692.22008
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