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Reducibility of induced representations for \(Sp(2n)\) and \(SO(n)\). (English) Zbl 0851.22021
Let \(G\) 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)]).
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
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