## 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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