Reducibility of induced representations for \(Sp(2n)\) and \(SO(n)\).

*(English)*Zbl 0851.22021Let \(G\) be one of the split \(p\)-adic groups \(\text{SO} (2n+1)\), \(\text{Sp} (2n)\), or \(\text{SO} (2n)\). A Levi subgroup of \(G\) is then a product of general linear groups and a smaller group of the same type as \(G\); a parabolic subgroup is “basic” if there is only one general linear group in the product for its Levi. The author shows that for \(G = \text{SO} (2n+1)\) or \(\text{Sp} (2n)\), the \(R\)-group decomposes into a product related to the product decomposition of the corresponding Levi subgroup. The same is sometimes true for \(\text{SO} (2n)\), but there are also some more complicated cases. This structure leads to a reduction of the computation of \(R\)-groups to the case of basic parabolic subgroups.

The paper includes a section discussing reducibility criteria for maximal parabolic subgroups in some cases, some remarks about the real case, and a proof of “multiplicity one” for representations induced from generic discrete series representations (the genericity condition was subsequently removed by R. Herb [Pac. J. Math. 161, 347-358 (1993; Zbl 0797.22007)]).

The paper includes a section discussing reducibility criteria for maximal parabolic subgroups in some cases, some remarks about the real case, and a proof of “multiplicity one” for representations induced from generic discrete series representations (the genericity condition was subsequently removed by R. Herb [Pac. J. Math. 161, 347-358 (1993; Zbl 0797.22007)]).

Reviewer: J.Repka (Toronto)

##### MSC:

22E50 | Representations of Lie and linear algebraic groups over local fields |

22E46 | Semisimple Lie groups and their representations |

20G05 | Representation theory for linear algebraic groups |