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Local coefficient matrices of metaplectic groups. (English) Zbl 1103.22009

The principal series representations of the \(n\)-fold metaplectic covers of \(G_r=GL_r(\mathbb F)\), where \(\mathbb F\) is a non-Archimedean local field, were described by D. A. Kazhdan and S. J. Patterson [Publ. Math., Inst. Hautes Étud. Sci. 59, 35-142 (1984; Zbl 0559.10026)]. While such representations do not, in general, have unique Whittaker models, there are finite-dimensional spaces of Whittaker functionals. The “local coefficient matrices” considered in the paper under review are formed by considering the action of the standard intertwining operator on a certain canonical basis of the functionals. In the case \(r=2\) the nonsingularity of local coefficient matrices is shown to be equivalent to the irreducibility of a particular class of representations in the unramified principal series. By considering these representations over different coverings of \(G_2\), it is then shown that a generalization of the local Shimura correspondence preserves irreducibility. For earlier results in this direction see Y. Flicker [Invent. Math. 57, 119-182 (1980; Zbl 0431.10014)].

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11F32 Modular correspondences, etc.
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F85 \(p\)-adic theory, local fields
22D30 Induced representations for locally compact groups
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