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Steinberg representation of \(\mathrm{GSp}(4)\): Bessel models and integral representation of \(L\)-functions. (English) Zbl 1264.11038

Author’s abstract: We obtain explicit formulas for the test vector in the Bessel model, and derive the criteria for existence and uniqueness of Bessel models for the unramified quadratic twists of the Steinberg representation \(\pi \) of \(\mathrm{GSp}_4(F)\), where \(F\) is a nonarchimedean local field of characteristic zero. We also give precise criteria for the Iwahori spherical vector in \(\pi \) to be a test vector. We apply the formulas for the test vector to obtain an integral representation of the local \(L\)-function of \(\pi \), twisted by any irreducible admissible representation of \(\mathrm{GL}_2(F)\). Using results of M. Furusawa [J. Reine Angew. Math. 438, 187–218 (1993; Zbl 0770.11025)] and of the author and R. Schmidt [Int. Math. Res. Not. 2009, No. 7, 1159–1212 (2009; Zbl 1244.11055)], we derive from this an integral representation for the global \(L\)-function of the irreducible cuspidal automorphic representation of \(\mathrm{GSp}_4(\mathbb A)\) obtained from a Siegel cuspidal Hecke newform, with respect to a Borel congruence subgroup of square-free level, twisted by any irreducible cuspidal automorphic representation of \(\mathrm{GL}_2(\mathbb A)\). A special-value result for this \(L\)-function, in the spirit of Deligne’s conjecture, is obtained.

MSC:

11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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