## Nagoya Mathematical Journal

### Doubling zeta integrals and local factors for metaplectic groups

Wee Teck Gan

#### Abstract

We develop the theory of the doubling zeta integral of Piatetski-Shapiro and Rallis for metaplectic groups $\operatorname{Mp}_{2n}$, and we use it to give precise definitions of the local $\gamma$-factors, $L$-factors, and $\epsilon$-factors for irreducible representations of $\operatorname{Mp}_{2n}\times\operatorname{GL}_{1}$, following the footsteps of Lapid and Rallis.

#### Article information

Source
Nagoya Math. J., Volume 208 (2012), 67-95.

Dates
First available in Project Euclid: 5 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1354716557

Digital Object Identifier
doi:10.1215/00277630-1815213

Mathematical Reviews number (MathSciNet)
MR3006697

Zentralblatt MATH identifier
1280.11028

#### Citation

Gan, Wee Teck. Doubling zeta integrals and local factors for metaplectic groups. Nagoya Math. J. 208 (2012), 67--95. doi:10.1215/00277630-1815213. https://projecteuclid.org/euclid.nmj/1354716557

#### References

• [GS] W. T. Gan and G. Savin, Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence, to appear in Compos. Math., available at http://www.math.nus.edu.sg/~matgw/metaplectic-1.pdf (accessed 15 August 2012).
• [GPSR] S. Gelbart, I. Piatetski-Shapiro, and S. Rallis, Explicit Constructions of Automorphic L-Functions, Lecture Notes in Math. 1254, Springer, Berlin, 1987.
• [LR] E. Lapid and S. Rallis, “On the local factors of representations of classical groups” in Automorphic Representations, $L$-Functions and Applications: Progress and Prospects, Ohio State Univ. Math. Res. Inst. Publ. 11, de Gruyter, Berlin, 2005, 309–359.
• [Li] J.-S. Li, Nonvanishing theorems for the cohomology of certain arithmetic quotients, J. Reine Angew. Math. 428 (1992), 177–217.
• [MVW] C. Moeglin, M.-F. Vigneras, and J.-L. Waldspurger, Correspondances de Howe sur un corps $p$-adique, Lecture Notes in Math. 1291, Springer, Berlin, 1987.
• [MW] C. Moeglin and J.-L. Waldspurger, Spectral Decomposition and Eisenstein Series, Cambridge Tracts in Math. 113, Cambridge University Press, Cambridge, 1995.
• [PSR] I. Piatetski-Shapiro and S. Rallis, $\epsilon$ factor of representations of classical groups, Proc. Natl. Acad. Sci. USA 83 (1986), 4589–4593.
• [R] R. Rao, On some explicit formulas in the theory of the Weil representation, Pacific J. Math. 157 (1993), 335–371.
• [Sun] B. Y. Sun, Dual pairs and contragredients of irreducible representations, Pacific J. Math. 249 (2011), 485–494.
• [Sw] J. Sweet, Functional equations of p-adic zeta integrals and representations of the metaplectic group, preprint, 1995.
• [Sz] D. Szpruch, The Langlands-Shahidi method for the metaplectic group and applications, preprint, arXiv:1004.3516v1 [math.NT]
• [Y] S. Yamana, $L$-functions and theta correspondences for classical groups, preprint, 2011.
• [Z] C. Zorn, Theta dichotomy and doubling epsilon factors for $\widetilde{\mathrm{SP}_{n}}(F)$, Amer. J. Math. 133 (2011), 1313–1364.