The unitary spectrum for real rank one groups.

*(English)*Zbl 0561.22009A 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.

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 |

##### Keywords:

Zuckermann derived functor modules; unitary representations of semisimple Lie groups of real rank one; \(L^ 2\)-index of the Dirac operator
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\textit{M. W. Baldoni Silva} and \textit{D. Barbasch}, Invent. Math. 72, 27--55 (1983; Zbl 0561.22009)

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##### References:

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