## Duke Mathematical Journal

### The twisted symmetric square $L$-function of $\operatorname {GL}(r)$

Shuichiro Takeda

#### Abstract

In this paper, we consider the (partial) symmetric square $L$-function $L^{S}(s,\pi,\operatorname {Sym}^{2}\otimes\chi)$ of an irreducible cuspidal automorphic representation $\pi$ of $\operatorname {GL}_{r}(\mathbb {A})$ twisted by a Hecke character $\chi$. In particular, we will show that the $L$-function $L^{S}(s,\pi,\operatorname {Sym}^{2}\otimes\chi)$ is holomorphic for the region $\operatorname {Re}(s)\gt 1-1/r$ with the exception that, if $\chi^{r}\omega^{2}=1$, a pole might occur at $s=1$, where $\omega$ is the central character of $\pi$. Our method of proof is essentially a (nontrivial) modification of the one by Bump and Ginzburg in which they considered the case $\chi=1$.

#### Article information

Source
Duke Math. J., Volume 163, Number 1 (2014), 175-266.

Dates
First available in Project Euclid: 8 January 2014

https://projecteuclid.org/euclid.dmj/1389190327

Digital Object Identifier
doi:10.1215/00127094-2405497

Mathematical Reviews number (MathSciNet)
MR3161314

Zentralblatt MATH identifier
1316.11037

#### Citation

Takeda, Shuichiro. The twisted symmetric square $L$ -function of $\operatorname {GL}(r)$. Duke Math. J. 163 (2014), no. 1, 175--266. doi:10.1215/00127094-2405497. https://projecteuclid.org/euclid.dmj/1389190327

#### References

• [1] J. Adams, Extensions of tori in ${\operatorname{SL} }(2)$, Pacific J. Math. 200 (2001), 257–271.
• [2] M. Asgari and F. Shahidi, Generic transfer for general spin groups, Duke Math. J. 132 (2006), 137–190.
• [3] M. Asgari and F. Shahidi, Image of functoriality for general spin groups, preprint https://www.math.okstate.edu/~asgari/res.html (accessed 26 November 2013).
• [4] D. Ban and C. Jantzen, The Langlands quotient theorem for finite extensions of p-adic groups, preprint, http://lagrange.math.siu.edu/ban/papersBan.htm (accessed 26 November 2013).
• [5] W. D. Banks, Twisted symmetric-square $L$-functions and the nonexistence of Siegel zeros on ${\operatorname{GL} }(3)$, Duke Math. J. 87 (1997), 343–353.
• [6] W. D. Banks, Exceptional representations of the metaplectic group, PhD dissertation, Stanford University, Stanford, Calif., 1994.
• [7] W. D. Banks, J. Levy, and M. R. Sepanski, Block-compatible metaplectic cocycles, J. Reine Angew. Math. 507 (1999), 131–163.
• [8] A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, 2nd ed., Math. Surveys Monogr. 67, Amer. Math. Soc., Providence, 2000.
• [9] D. Bump, Automorphic Forms and Representations, Cambridge Stud. Adv. Math. 55, Cambridge Univ. Press, Cambridge, 1997.
• [10] D. Bump and D. Ginzburg, Symmetric square $L$-functions on ${\operatorname{GL} }(r)$, Ann. of Math. (2) 136 (1992), 137–205.
• [11] Y. Z. Flicker, Automorphic forms on covering groups of ${\operatorname{GL} }(2)$, Invent. Math. 57 (1980), 119–182.
• [12] Y. Z. Flicker and D. A. Kazhdan, Metaplectic correspondence, Inst. Hautes Études Sci. Publ. Math. 64 (1986), 53–110.
• [13] S. Gelbart, Weil’s Representation and the Spectrum of the Metaplectic Group, Lecture Notes in Math. 530, Springer, Berlin, 1976.
• [14] S. Gelbart, and H. Jacquet, A relation between automorphic representations of ${\operatorname{GL} }(2)$ and ${\operatorname{GL} }(3)$, Ann. Sci. Ecole Norm. Sup. (4) 11 (1978), 471–542.
• [15] S. Gelbart and I. I. Piatetski-Shapiro, Distinguished representations and modular forms of half-integral weight, Invent. Math. 59 (1980), 145–188.
• [16] M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, with an appendix by V. G. Berkovich, Ann. of Math. Stud. 151, Princeton Univ. Press, Princeton, N.J., 2001.
• [17] G. Henniart, Une preuve simple des conjectures de Langlands pour ${\operatorname{GL} }(n)$ sur un corps $p$-adique, Invent. Math. 139 (2000), 439–455.
• [18] H. Jacquet and J. Shalika, “Exterior square $L$-functions” in Automorphic Forms, Shimura Varieties, and $L$-functions, Vol. II (Ann Arbor, Mich., 1988), Perspect. Math. 11, Academic Press, Boston, 1990, 143–226.
• [19] A. C. Kable, Exceptional representations of the metaplectic double cover of the general linear group, PhD dissertation, Oklahoma State University, Stillwater, Okla., 1997.
• [20] A. C. Kable, The tensor product of exceptional representations on the general linear group, Ann. Sci. École Norm. Sup. (4) 34 (2001), 741–769.
• [21] D. A. Kazhdan and S. J. Patterson, Metaplectic forms, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 35–142.
• [22] T. Kubota, On Automorphic Functions and the Reciprocity Law in a Number Field, Department of Mathematics, Kyoto University, Lectures in Mathematics 2, Kinokuniya, Tokyo, 1969.
• [23] R. P. Langlands, “The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups” in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo.), Amer. Math. Soc., Providence, 1965, 143–148.
• [24] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Math. Monogr., Oxford Univ. Press, New York, 1979.
• [25] H. Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. 2 (1969), 1–62.
• [26] P. Mezo, Metaplectic tensor products for irreducible representations, Pacific J. Math. 215 (2004), 85–96.
• [27] C. Moeglin and J.-L. Waldspurger, Spectral Decomposition and Eisenstein Series, Cambridge Tracts in Math. 113, Cambridge Univ. Press, Cambridge, 1995.
• [28] S. J. Patterson and I. I. Piatetski-Shapiro, The symmetric-square $L$-function attached to a cuspidal automorphic representation of ${\operatorname{GL} }_{3}$, Math. Ann. 283 (1989), 551–572.
• [29] R. Ranga Rao, On some explicit formulas in the theory of Weil representation, Pacific J. Math. 157 (1993), 335–371.
• [30] F. Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$-adic groups, Ann. of Math. (2) 132 (1990), 273–330.
• [31] F. Shahidi, On non-vanishing of twisted symmetric and exterior square $L$-functions for ${\operatorname{GL} }(n)$, Pacific J. Math. 181 (1997), 311–322.
• [32] J. Shalika, The multiplicity one theorem for ${\operatorname{GL} }_{n}$, Ann. of Math. (2) 100 (1974), 171–193.
• [33] G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31 (1975), 79–98.
• [34] T. Shintani, On an explicit formula for class-$1$ “Whittaker functions” on $\operatorname{GL} (n)$ over $P$-adic fields, Proc. Japan Acad. 52 (1976), 180–182.
• [35] S. Takeda, Metaplectic tensor products for automorphic representations of ${\widetilde{\operatorname {GL}} }(r)$, preprint, http://www.math.missouri.edu/~takedas/research.html (accessed 26 November 2013).
• [36] S. Takeda, On a certain metaplectic Eisenstein series and the twisted symmetric square $L$-function, preprint, http://www.math.missouri.edu/~takedas/research.html (accessed 26 November 2013).
• [37] J.-L. Waldspurger, Correspondance de Shimura, J. Math. Pures Appl. 59 (1980), 1–132.
• [38] J.-L. Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), 219–307.
• [39] N. R. Wallach, “Asymptotic expansions of generalized matrix entries of representations of real reductive groups” in Lie Group Representations, I (College Park, Md., 1982/1983), Lecture Notes in Math. 1024, Springer, Berlin, 1983, 287–369.
• [40] A. Weil, Sur certains groupes d’operateurs unitaires, Acta Math. 111 (1964) 143–211.