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Ramified degenerate principal series representations for \(Sp(n)\). (English) Zbl 0787.22019
In approaching a general Weil-Siegel formula in the divergent range a central role is played by the degenerate principal series representations of the symplectic group. Let \(G=Sp_ n(F)\) be the symplectic group over a non-archimedean local field \(F\) of characteristic zero. The group \(G\) has a maximal parabolic subgroup of the form \(P=MN\) with Levi factor \(M\cong GL_ n(F)\) and unipotent radical \(N=Sym_ n(F)\). For any unitary character \(\chi\) of \(F^*\) and for any \(s\in\mathbb{C}\), there is the representation \(I(s,\chi)= \text{Ind}_ P^ G \chi\cdot|\;|^ s\), where the induction is normalized so that \(I(s,\chi)\) is naturally unitarizable when \(s\) is pure imaginary. Provided the character \(\chi\) is unramified, the points of reducibility and a complete description of the constituents and composition series was given by R. Gustafson [The degenerate principal series of \(Sp(2n)\) (Mem. Am. Math. Soc. 248) (1981; Zbl 0482.22013)]. These methods cannot be applied in the ramified case.
In view of the global applications to an extension of the Siegel-Weil formula, the case of an arbitrary character is dealt with in the paper under review. A complete description of the points of reducibility, components and composition series is given. It is shown that all of the reducibility is accounted for by submodules coming from the Weil representation associated to quadratic forms over \(F\).

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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