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Generalized Whittaker functions on \(\text{GSp}(2, \mathbb R)\) associated with indefinite quadratic forms. (English) Zbl 1268.22018

Summary: We study the generalized Whittaker models for \(G = \text{GSp}(2, \mathbb R)\) associated with indefinite binary quadratic forms when they arise from two standard representations of \(G\): (i) a generalized principal series representation induced from the non-Siegel maximal parabolic subgroup and (ii) a (limit of) large discrete series representation. We prove the uniqueness of such models with moderate growth property. Moreover we express the values of the corresponding generalized Whittaker functions on a one-parameter subgroup of \(G\) in terms of the Meijer \(G\)-functions.

MSC:

22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E30 Analysis on real and complex Lie groups
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