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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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