On the unitary dual of the classical Lie groups. II: Representations of \(SO(n,m)\) inside the dominant Weyl chamber.

*(English)*Zbl 0782.22010There is the conjecture due to Vogan that a unitary representation of a real reductive Lie group can be obtained by cohomological parabolic induction from a special unipotent representation of a subgroup. If the integral infinitesimal character is regular then the special unipotent representations involved are one dimensional, and the conjecture says: Suppose \(X\) is an irreducible unitary Harish-Chandra module of \(G\) whose infinitesimal character minus half the sum of the positive roots is dominant. Then there are a \(\theta\)-stable parabolic subalgebra \(\mathfrak q\) of the complexified Lie algebra \(\mathfrak g\) and a unitary one dimensional character \(\lambda\) of the Levi subgroup \(L\) of \(\mathfrak q\) such that \(X\) is isomorphic to the Zuckerman module \(R^{\mathfrak g}_{\mathfrak q}(\mathbb{C}_ \lambda)\).

In part I [ibid. 68, 251-303 (1988; Zbl 0692.22007)] this conjecture was proved when \(G\) is \(SL_ n(\mathbb{R})\), \(Sp_ n(\mathbb{R})\) or \(SU(p,q)\). The paper under review deals with the case \(SO(n,m)\). The proof is by reduction to a subgroup of \(G\) of smaller dimension. The main tool used is the Dirac operator inequality as e.g., stated in Part I.

In part I [ibid. 68, 251-303 (1988; Zbl 0692.22007)] this conjecture was proved when \(G\) is \(SL_ n(\mathbb{R})\), \(Sp_ n(\mathbb{R})\) or \(SU(p,q)\). The paper under review deals with the case \(SO(n,m)\). The proof is by reduction to a subgroup of \(G\) of smaller dimension. The main tool used is the Dirac operator inequality as e.g., stated in Part I.

Reviewer: J.Schwermer (Eichstätt)

##### MSC:

22E46 | Semisimple Lie groups and their representations |

22E30 | Analysis on real and complex Lie groups |

22E15 | General properties and structure of real Lie groups |

##### Keywords:

unitary representation; real reductive Lie group; cohomological parabolic induction; special unipotent representations; irreducible unitary Harish- Chandra module; positive roots; complexified Lie algebra; Zuckerman module; Dirac operator inequality##### References:

[1] | A. Borel and N. Wallach : Continuous cohomology, discrete subgroups and representations of reductive subgroups , in Annals of Mathematics Studies Vol. 94, Princeton University Press, 1980. · Zbl 0443.22010 |

[2] | S. Salamanca-Riba : On the unitary dual of some classical Lie groups , Compositio Math. 68 (1988), 251-303. · Zbl 0692.22007 · numdam:CM_1988__68_3_251_0 · eudml:89938 |

[3] | B. Speh and D. Vogan : Reducibility of generalized principal series representations , Acta Math. 145 (1980), 227-229. · Zbl 0457.22011 · doi:10.1007/BF02414191 |

[4] | D. Vogan : Representations of Real Reductive Lie Groups , Birkhäuser, Boston-Basel- Stuttgart, 1981. · Zbl 0469.22012 |

[5] | D. Vogan : Unitarizability of certain series of representations , Annals Math. 120 (1984),141-187. · Zbl 0561.22010 · doi:10.2307/2007074 |

[6] | G. Zuckerman : On Construction of Representations by Derived Functors . Handwritten notes, 1977. |

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