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Doubling zeta integrals and local factors for metaplectic groups. (English) Zbl 1280.11028

The purpose of the paper under review is to extend a theory of doubling zeta integral to the case of metaplectic group \(\mathrm{Mp}(2n)\), the unique non-trivial two-fold cover of the symplectic group \(\mathrm{Sp}(2n)\). This theory was introduced by I. I. Piatetski-Shapiro and S. Rallis [Proc. Natl. Acad. Sci. USA 83, No. 13, 4589–4593 (1986; Zbl 0599.12012) and further studied by E. M. Lapid and S. Rallis [Ohio State Univ. Math. Res. Inst. Publ. 11, 309–359 (2005; Zbl 1188.11023)] in the case of symplectic, orthogonal and unitary groups.
Let \(k\) denote a local field of characteristic zero and let \(\psi\) be an additive character of \(k\). For an irreducible genuine representation \(\sigma \otimes \chi\) of \(\mathrm{Mp}(2n) \times \mathrm{GL}(1, k)\), the author defines the local \(\gamma\)-factor \(\gamma (s, \sigma \times \chi, \psi)\), the local \(L\)-factor \(L(s, \sigma, \chi, \psi)\) and the local \(\varepsilon\)-factor \(\varepsilon(s, \sigma \times \chi, \psi)\), following the work of Lapid and Rallis. Further, the author verifies that the local \(\gamma\)-factors satisfy certain properties which characterize them uniquely.
We emphasize two main differences between the metaplectic and the linear case:
First, in the metaplectic case the local \(L\)-factor depends on the character \(\psi\), while in the linear case it does not.
Second, in the metaplectic case the local factors change when the representation \(\sigma\) is replaced by its contragredient, while in the symplectic and orthogonal case the local factors attached to an irreducible representation \(\sigma \otimes \chi\), where \(\sigma\) is a representation of either symplectic or orthogonal group, do not change when \(\sigma\) is replaced by its contragredient.

MSC:

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

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