# On Generalized Fourier Transforms for Standard *L*-Functions

## Abstract

Any generalization of the method of Godement–Jacquet on principal *L*-functions for *GL*(*n*) to other groups as perceived by Braverman–Kazhdan/Ngo requires a Fourier transform on a space of Schwartz functions. In the case of standard *L*-functions for classical groups, a theory of this nature was developed by Piatetski-Shapiro and Rallis, called the doubling method. It was later that Braverman and Kazhdan, using an algebro-geometric approach, different from doubling method, introduced a space of Schwartz functions and a Fourier transform, which projected onto those from doubling method. In both methods a normalized intertwining operator played the role of the Fourier transform. The purpose of this paper is to show that the Fourier transform of Braverman–Kazhdan projects onto that of doubling method. In particular, we show that they preserve their corresponding basic functions. The normalizations involved are not the standard ones suggested by Langlands, but rather a singular version of local coefficients of Langlands–Shahidi method. The basic function will require a shift by 1/2 as dictated by doubling construction, reflecting the global theory, and begs explanation when compared with the work of Bouthier–Ngo–Sakellaridis. This matter is further discussed in an appendix by Wen-Wei Li.

## Keywords

Braverman-Kazhdan Schwartz spaces and Fourier transforms Doubling method Normalized intertwining operators## Notes

### Acknowledgements

Confusions stemming from the shift \(s+\frac 12\), which did not seem to agree with Ngo’s shift [N3, BNS], did result in a number of communications with Erez Lapid for which I like to thank him. Similar gratitude is owed to Dihua Jiang, Wee Teck Gan, Shunsuke Yamana, David Kazhdan, and Jayce Getz. I also like to thank Werner Müller, Sug Woo Shin, and Nicolas Templier for their invitation to the Simons Symposium at Elmau, Germany, in April of 2016 and for the present proceedings. Parts of this paper were presented as a series of lectures at University of Minnesota where author was invited as an Ordway Distinguished Visitor during the Fall of 2016 and for which thanks are due to Dihua Jiang. Last but not least, I like to thank Wen-Wei Li for a numbers of helpful comments and communications after the first version of this manuscript was distributed, which in particular led to his insightful appendix [Li2] to this paper.

## References

- [A]J. Arthur,
*Intertwining operators and residues I. Weighted characters*, J. Funct. Anal.,**84**(1989), 19–84.MathSciNetCrossRefGoogle Scholar - [BGKP]A. Braverman, H. Garland, D. Kazhdan, and M. Patnaik,
*An affine Gindikin–Karpelevic formula, in Perspectives in Representation Theory*, Contemp. Math. Vol. 610, AMS, 43–64 (2014).Google Scholar - [BNS]A. Bouthier, B.C. Ngô, Y. Sakellaridis,
*On the formal arc space of a reductive monoid*, American J. Math., Igusa Memorial Issue, 2016, 138, 1: 81–108.MathSciNetCrossRefGoogle Scholar - [BK1]A. Braverman and D. Kazhdan,
*γ-functions of representations and lifting. Geom. Funct. Anal.*, (Special Volume, Part I):237–278, 2000. With an appendix by V. Vologodsky, GAFA 2000 (Tel Aviv, 1999).Google Scholar - [BK2],
*Normalized intertwining operators and nilpotent elements in the Langlands dual group*, Moscow Math. J. 2 (2002), no. 3, 533–553.Google Scholar - [BK3],
*On the Schwartz space of the basic affine space, Selecta Math. (N.S.),*5(1):1–28, (1999).Google Scholar - [C]R. W. Carter,
*Finite Groups of Lie Type, Conjugacy Classes and Complex Characters*, Wiley Classics Library, John Wiley & Sons Ltd, 1993.Google Scholar - [ChN]S. Cheng and B.C. Ngô,
*On a conjecture of Braverman and Kazhdan*, Duke Math. J., China, 200–256 (2017).Google Scholar - [FG]S. Friedberg and D. Goldberg,
*On local coefficients for non-generic representations of some classical groups*, Comp. Math.**116**(1999), no. 2,, 133–166.MathSciNetCrossRefGoogle Scholar - [Gan]W. T. Gan,
*Doubling zeta integrals and local factors for metaplectic groups*, Nagoya Math. J.**208**(2012), 67–95.MathSciNetCrossRefGoogle Scholar - [GPSR]Stephen Gelbart, Ilya Piatetski–Shapiro, and Stephen Rallis,
*Explicit constructions of automorphic**L-functions*, volume 1254 of*Lecture Notes in Mathematics*, Springer-Verlag, Berlin, 1987.Google Scholar - [GL]J. Getz and B. Liu,
*A refined Poisson summation formula for certain Braverman–Kazhdan spaces*, Preprint.Google Scholar - [GJ]R. Godement and H. Jacquet,
*Zeta functions of simple algebras*, Lecture Notes in Mathematics, vo. 260, Springer-Verlag, Berlin-New York, 1972.Google Scholar - [JS1]H. Jacquet and J. A. Shalika,
*On Euler products and the classification of automorphic representations, I*, Amer. J. Math.**103**:3 (1981), 499–558.MathSciNetCrossRefGoogle Scholar - [JS2],
*Exterior square**L-functions, in Automorphic Forms, Shimura Varieties, and**L-functions*, Vol. 2, ed. L. Clozel and J. S. Milne, Perspect. Math. 11, Academic Press, Boston (1990), 143–226.Google Scholar - [Ka]M. L. Karel,
*Functional equations of Whittaker functions on p-adic groups*, Amer. J. Math.**101**:6 (1979), 1303–1325.MathSciNetCrossRefGoogle Scholar - [La]R.P. Langlands,
*On the Functional Equations Satisfied by Eisenstein Series*, Lecture Notes in Math., Vol 544, Springer-Verlag, Berlin-Heidelberg-New york, 1976.CrossRefGoogle Scholar - [LR]E. M. Lapid and S. Rallis,
*On the local factors of representations of classical groups*, in*Automorphic representations, L-functions and Applications: Progress and Prospects*, edited by J. W. Cogdell et al., Ohio State Univ. Math. Res. Inst. Publ.**11**; de Gruyter, Berlin, 2005, pp. 309–359.Google Scholar - [L]J.-S. Li,,
*Singular unitary representations of classical groups, Invent. Math*.**97**:*2*(1989), 237–255.Google Scholar - [Li]Wen-Wei Li,
*Zeta integrals, Schwartz spaces and local functional equations*, Preprint, 2015.Google Scholar - [Li2],
*A comparison of basic functions*Appendix to this paper.Google Scholar - [N1]B.C. Ngô, On a certain sum of automorphic
*L*-functions. In*Automorphic Forms and Related Geometry: Assessing the Legacy of I.I. Piatetski-Shapiro*, volume 614 of*Contemp. Math.*, pages 337–343. Amer. Math. Soc., Providence, RI, 2014.Google Scholar - [N2],
*Semi-group and basic functions, Letter to Sakellaridis*.Google Scholar - [N3],
*Geometry of arc spaces, generalized Hankel transforms and Langlands functoriality*, preprint, 2016.Google Scholar - [P]M. Patnaik,
*Unramified Whittaker functions on p-adic loop groups*, Amer. J. Math.,**139**(2017), 175–215.MathSciNetCrossRefGoogle Scholar - [PSR]I. Piatetski–Shapiro, and S. Rallis,
*ε**factor of representations of classical groups*, Proc. Nat. Acad. Sci. U.S.A.**83(13)**:, 4589–4593, 1986.Google Scholar - [Sh1]F. Shahidi,
*On certain L-functions*, Amer. J. Math.,**103**(1981), 297–355.MathSciNetCrossRefGoogle Scholar - [Sh2],
*Local coefficients as Artin factors for real groups*, Duke Math. J.,**52**(1985), 973–1007.MathSciNetCrossRefGoogle Scholar - [Sh3],
*A proof of Langlands conjecture on Plancherel measures; Complementary series for p-adic groups*, Annals of Math.,**132**(1990), 273–330.Google Scholar - [Sh4],
*Local Factors, Reciprocity and Vinberg Monoids, in “Prime Numbers and Representation Theory”*, Lecture Series of Modern Number Theory, Vol. 2, Science Press, Beijing, 2017. ISBN: 9787030533401.Google Scholar - [V]E.B. Vinberg, On reductive algebraic semigroups. In
*Lie groups and Lie algebras: E.B. Dynkin’s Seminar*, volume 169 of*Amer. Math. Soc. Transl. Ser. 2*, Amer. Math. Soc., Providence, RI, 1995.Google Scholar - [Y1]S. Yamana,
*L–functions and theta correspondence for classical groups, Invent. Math.***196**(2014), 651–732.MathSciNetGoogle Scholar - [Y2],
*The Siegel–Weil formula for unitary groups*, Pacific J. Math.**264**(2013), 235–257.MathSciNetCrossRefGoogle Scholar