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Degenerate principal series for orthogonal groups. (English) Zbl 0776.22004

Let \(F\) be a \(p\)-adic field, \(G=SO_ n(F)\), \(P=MU\) a maximal proper parabolic subgroup of \(G\), and \(\sigma\) a one-dimensional representation of \(M\). Consider the induced representation \(\pi=\text{Ind}^ G_ P \sigma\otimes 1\). Using a technique of Tadić involving Jacquet modules, we analyze the reducibility of \(\pi\). In particular, we determine which values of \(\sigma\) make \(\pi\) reducible and determine Langlands data and Jacquet modules for the components of \(\pi\) when it is reducible. We do this in the case where \(\sigma\) satisfies a regularity condition (for all \(n\)), and in general for \(n\leq 7\).

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
22D30 Induced representations for locally compact groups
22E35 Analysis on \(p\)-adic Lie groups
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