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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 \(n\).

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