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 \(n\).

Reviewer: S.Gelbart (Rehovot)

##### MSC:

22E50 | Representations of Lie and linear algebraic groups over local fields |

22D30 | Induced representations for locally compact groups |