The twisted symmetric square \(L\)-function of \(\mathrm{GL}(r)\). (English) Zbl 1316.11037

Takeda, Shuichiro

Duke Math. J. 163, No. 1, 175-266 (2014).

For the symmetric square map
\[ \text{Sym}^2 : \mathrm{GL}_r({\mathbb C}) \rightarrow \mathrm{GL}_{\frac {1}{2}r(r+1)} ({\mathbb C}), \] if we consider the twisted symmetric square \(L\)-function \(L(s,\pi, \text{Sym}^2 \otimes \chi)\) where \(\pi\) is an irreducible cuspidal representation of \(\mathrm{GL}_r({\mathbb A})\), \({\mathbb A}\) is the ring of adeles over a number field \(F\) and \(\chi\) is a unitary Hecke character on \({\mathbb A}^{\times}\), it is known that this \(L\)-function admits meromorphic continuation and satisfies a functional equation by Langlands-Shahidi method (see theorem 7.7 of [F. Shahidi, Ann. Math. (2) 132, No. 2, 273–330 (1990; Zbl 0780.22005)]). However this method is unable to determine the locations of the possible poles of \(L(s,\pi, \text{Sym}^2 \otimes \chi)\). The main theme of the paper under review is to determine them to some extent, though it is done only for the partial \(L\)-function \(L^S(s,\pi, \text{Sym}^2 \otimes \chi)\). More specifically, let \(S\) be the finite set of places that contain all the Archimedean places and non-Archimedean places where \(\pi\) or \(\chi\) ramifies. For \(v \notin S\), each local \(\pi_v\) is parametrized by the Satake parameters, a set of \(r\) complex numbers \(\{ \alpha_{v,1}, \cdots \alpha_{v,r}\}\). Then we have,
\[ L_v (s, \pi_v, \text{Sym}^2 \otimes \chi_v) = \prod \limits_{i \leq j} \frac {1}{(1-\chi_v(\overline {\omega_v}) \alpha_{v,i} \alpha_{v,j} q_v^{-s})} \] where \(\overline {\omega_v}\) is the uniformizer of \(F_v\) and \(q_v\) is the order of the residue field. Set
\[ L^S (s, \pi, \text{Sym}^2 \otimes \chi) = \prod \limits_{v \notin S} L_v (s, \pi_v, \text{Sym}^2 \otimes \chi_v). \] It is established that : If \(\pi\) be a cuspidal automorphic representation of \(\mathrm{GL}_r({\mathbb A})\) with unitary central character \(\omega\) and \(\chi\) being a unitary Hecke character, then for each Archimedean \(v\), there exists an integer \(N_v \geq 0\) such that the product \[ L^S (s, \pi, \text{Sym}^2 \otimes \chi) \prod \limits_{v | \infty} L_v (rs-r+1, \chi_v^r \omega_v^2)^{-N_v} \] is holomorphic everywhere except at \(s=0\) and \(s=1\). Moreover, there is no pole if \(\chi^r \omega^2 \neq 1\).
As a corollary, it is derived that the incomplete twisted \(L\)-function \(L^S (s, \pi, \text{Sym}^2 \otimes \chi)\) is holomorphic everywhere in the region \(\mathrm{Re} (s) > 1-\frac {1}{2r}\) except at \(s=1\). Moreover, there is no pole at \(s=1\) if \(\chi^r \omega^2 \neq 1\).
For more related works to the paper, the readers are referred to for e.g., [S. Gelbart and H. Jacquet, Ann. Sci. Éc. Norm. Supér. (4) 11, No. 4, 471–542 (1978; Zbl 0406.10022); G. Shimura, Proc. Lond. Math. Soc. (3) 31, 79–98 (1975; Zbl 0311.10029); S. J. Patterson and I. I. Piatetski-Shapiro, Math. Ann. 283, No. 4, 551–572 (1989; Zbl 0645.22007); D. Bump and D. Ginzburg, Ann. Math. (2) 136, No. 1, 137–205 (1992; Zbl 0753.11021); D. A. Kazhdan and S. J. Patterson, Publ. Math., Inst. Hautes Étud. Sci. 59, 35–142 (1984; Zbl 0559.10026); W. D. Banks, Exceptional representations of the metaplectic group. Stanford: Stanford University (PhD Thesis) (1994)] and [A. C. Kable, Exceptional representations of the metaplectic double cover of the general linear group. Stillwater: Oklahoma State University (1997)].

Reviewer: A. Sankaranarayanan (Mumbai)

MSC:

11F66

Langlands \(L\)-functions; one variable Dirichlet series and functional equations

Keywords:

twisted symmetric square \(L\)-function; cuspidal automorphic representation; Hecke character; metaplectic double cover; Rankin-Selberg integral

Citations:

Zbl 0406.10022; Zbl 0311.10029; Zbl 0645.22007; Zbl 0753.11021; Zbl 0559.10026; Zbl 0780.22005

References:

[1]	J. Adams, Extensions of tori in \({\operatorname{SL} }(2)\) , Pacific J. Math. 200 (2001), 257-271. · Zbl 1054.11027 · doi:10.2140/pjm.2001.200.257
[2]	M. Asgari and F. Shahidi, Generic transfer for general spin groups , Duke Math. J. 132 (2006), 137-190. · Zbl 1099.11028 · doi:10.1215/S0012-7094-06-13214-3
[3]	M. Asgari and F. Shahidi, Image of functoriality for general spin groups , preprint (accessed 26 November 2013). · Zbl 1298.11042
[4]	D. Ban and C. Jantzen, The Langlands quotient theorem for finite extensions of p-adic groups , preprint, (accessed 26 November 2013). · Zbl 1304.22019
[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. · Zbl 0880.11045 · doi:10.1215/S0012-7094-97-08713-5
[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. · Zbl 0914.20039 · doi:10.1515/crll.1999.507.131
[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. · Zbl 0980.22015
[9]	D. Bump, Automorphic Forms and Representations , Cambridge Stud. Adv. Math. 55 , Cambridge Univ. Press, Cambridge, 1997. · Zbl 0911.11022
[10]	D. Bump and D. Ginzburg, Symmetric square \(L\)-functions on \({\operatorname{GL} }(r)\) , Ann. of Math. (2) 136 (1992), 137-205. · Zbl 0753.11021 · doi:10.2307/2946548
[11]	Y. Z. Flicker, Automorphic forms on covering groups of \({\operatorname{GL} }(2)\) , Invent. Math. 57 (1980), 119-182. · Zbl 0431.10014 · doi:10.1007/BF01390092
[12]	Y. Z. Flicker and D. A. Kazhdan, Metaplectic correspondence , Inst. Hautes Études Sci. Publ. Math. 64 (1986), 53-110. · Zbl 0616.10024 · doi:10.1007/BF02699192
[13]	S. Gelbart, Weil’s Representation and the Spectrum of the Metaplectic Group , Lecture Notes in Math. 530 , Springer, Berlin, 1976. · Zbl 0365.22017 · doi:10.1007/BFb0077830
[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. · Zbl 0406.10022
[15]	S. Gelbart and I. I. Piatetski-Shapiro, Distinguished representations and modular forms of half-integral weight , Invent. Math. 59 (1980), 145-188. · Zbl 0426.10027 · doi:10.1007/BF01390042
[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. · Zbl 1036.11027
[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. · Zbl 1048.11092 · doi:10.1007/s002220050012
[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. · Zbl 0695.10025
[19]	A. C. Kable, Exceptional representations of the metaplectic double cover of the general linear group , PhD dissertation, Oklahoma State University, Stillwater, Okla., 1997. · Zbl 1185.11076
[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. · Zbl 1005.20033 · doi:10.1016/S0012-9593(01)01075-8
[21]	D. A. Kazhdan and S. J. Patterson, Metaplectic forms , Inst. Hautes Études Sci. Publ. Math. 59 (1984), 35-142. · Zbl 0559.10026 · doi:10.1007/BF02698770
[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. · Zbl 0231.10017
[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. · Zbl 0218.20041
[24]	I. G. Macdonald, Symmetric Functions and Hall Polynomials , Oxford Math. Monogr., Oxford Univ. Press, New York, 1979. · Zbl 0487.20007
[25]	H. Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés , Ann. Sci. École Norm. Sup. 2 (1969), 1-62. · Zbl 0261.20025
[26]	P. Mezo, Metaplectic tensor products for irreducible representations , Pacific J. Math. 215 (2004), 85-96. · Zbl 1068.22018 · doi:10.2140/pjm.2004.215.85
[27]	C. Moeglin and J.-L. Waldspurger, Spectral Decomposition and Eisenstein Series , Cambridge Tracts in Math. 113 , Cambridge Univ. Press, Cambridge, 1995. · Zbl 0846.11032
[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. · Zbl 0645.22007 · doi:10.1007/BF01442854
[29]	R. Ranga Rao, On some explicit formulas in the theory of Weil representation , Pacific J. Math. 157 (1993), 335-371. · Zbl 0794.58017 · doi:10.2140/pjm.1993.157.335
[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. · Zbl 0780.22005 · doi:10.2307/1971524
[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. · Zbl 1001.11021
[32]	J. Shalika, The multiplicity one theorem for \({\operatorname{GL} }_{n}\) , Ann. of Math. (2) 100 (1974), 171-193. · Zbl 0316.12010 · doi:10.2307/1971071
[33]	G. Shimura, On the holomorphy of certain Dirichlet series , Proc. London Math. Soc. (3) 31 (1975), 79-98. · Zbl 0311.10029 · doi:10.1112/plms/s3-31.1.79
[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. · Zbl 0387.43002 · doi:10.3792/pja/1195518347
[35]	S. Takeda, Metaplectic tensor products for automorphic representations of \({\widetilde{\operatorname {GL}} }(r)\) , preprint, (accessed 26 November 2013).
[36]	S. Takeda, On a certain metaplectic Eisenstein series and the twisted symmetric square \(L\)-function , preprint, (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. · Zbl 0724.11026 · doi:10.1515/form.1991.3.219
[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. · Zbl 0553.22005
[40]	A. Weil, Sur certains groupes d’operateurs unitaires , Acta Math. 111 (1964) 143-211. · Zbl 0203.03305 · doi:10.1007/BF02391012