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Degenerate principal series for symplectic groups. (English) Zbl 0814.22004
Mem. Am. Math. Soc. 488, 111 p. (1993).
Let $$F$$ be a $$p$$-adic field with $$\text{ch}(F) = 0$$ and $$G = \text{Sp}_{2n}(F)$$. Let $$P = MU$$ be a maximal parabolic subgroup of $$G$$, so $$M \cong \text{GL}_ k(F) \times \text{Sp}_{2(n - k)}(F)$$, for some $$k$$, $$1 \leq k \leq n$$. A quasi-character of $$M$$ is of the form $$\chi \circ \text{det}$$, $$\chi \in \widehat{F}^ \times$$, and may be extended trivially to $$P$$. In this paper, the author investigates the reducibility of the induced representation $$\pi = \text{Ind}^ G_ P \chi$$ (normalized so that unitary $$\chi$$ induces to unitary $$\pi$$). Such “degenerate” principal series representations were previously studied by several authors, notably R. Gustafson [for the case $$M \cong\text{GL}_ n(F)$$ and $$\chi$$ unramified; Mem. Am. Math. Soc. 248 (1981; Zbl 0482.22013)] and S. Kudla and S. Rallis [for the case of $$M \cong\text{GL}_ n(F)$$ and $$\chi$$ arbitrary; cfr. Isr. J. Math. 78, No. 2-3, 209-256 (1992; Zbl 0787.22019)]. Two different approaches are used in this paper: the first reduces the problem to the corresponding question about an associated finite-dimensional representation of a certain Hecke algebra; the second (is based on technique of Tadic’s and) involves an analysis of Jacquet modules. Fairly general results are given here for the case of $$P$$ with $$M \cong F^ \times\times \text{Sp}_{2(n-1)}(F)$$; for an arbitrary $$P$$, the method of Jacquet modules yields some general results, as well as specific ones for small $$n$$.

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 22D30 Induced representations for locally compact groups
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