## Constructing Weyl group multiple Dirichlet series

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- by Gautam Chinta and Paul E. Gunnells PDF
- J. Amer. Math. Soc.
**23**(2010), 189-215 Request permission

## Abstract:

Let $\Phi$ be a reduced root system of rank $r$. A*Weyl group multiple Dirichlet series*for $\Phi$ is a Dirichlet series in $r$ complex variables $s_1,\dots ,s_r$, initially converging for $\mathrm {Re}(s_i)$ sufficiently large, that has meromorphic continuation to ${\mathbb C}^r$ and satisfies functional equations under the transformations of ${\mathbb C}^r$ corresponding to the Weyl group of $\Phi$. A heuristic definition of such a series was given by Brubaker, Bump, Chinta, Friedberg, and Hoffstein, and they have been investigated in certain special cases by others. In this paper we generalize results by Chinta and Gunnells to construct Weyl group multiple Dirichlet series by a uniform method and show in all cases that they have the expected properties.

## References

- Ben Brubaker and Daniel Bump,
*On Kubota’s Dirichlet series*, J. Reine Angew. Math.**598**(2006), 159–184. MR**2270571**, DOI 10.1515/CRELLE.2006.073 - Benjamin Brubaker and Daniel Bump,
*Residues of Weyl group multiple Dirichlet series associated to $\widetilde \textrm {GL}_{n+1}$*, Multiple Dirichlet series, automorphic forms, and analytic number theory, Proc. Sympos. Pure Math., vol. 75, Amer. Math. Soc., Providence, RI, 2006, pp. 115–134. MR**2279933**, DOI 10.1090/pspum/075/2279933 - Benjamin Brubaker, Daniel Bump, Gautam Chinta, Solomon Friedberg, and Jeffrey Hoffstein,
*Weyl group multiple Dirichlet series. I*, Multiple Dirichlet series, automorphic forms, and analytic number theory, Proc. Sympos. Pure Math., vol. 75, Amer. Math. Soc., Providence, RI, 2006, pp. 91–114. MR**2279932**, DOI 10.1090/pspum/075/2279932 - B. Brubaker, D. Bump, and S. Friedberg. Weyl group multiple Dirichlet series, Eisenstein series and crystal bases. Submitted.
- B. Brubaker, D. Bump, and S. Friedberg. Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory. Submitted.
- Ben Brubaker, Daniel Bump, and Solomon Friedberg,
*Weyl group multiple Dirichlet series. II. The stable case*, Invent. Math.**165**(2006), no. 2, 325–355. MR**2231959**, DOI 10.1007/s00222-005-0496-2 - Ben Brubaker, Daniel Bump, and Solomon Friedberg,
*Twisted Weyl group multiple Dirichlet series: the stable case*, Eisenstein series and applications, Progr. Math., vol. 258, Birkhäuser Boston, Boston, MA, 2008, pp. 1–26. MR**2402679**, DOI 10.1007/978-0-8176-4639-4_{1} - B. Brubaker, D. Bump, S. Friedberg, and J. Hoffstein,
*Weyl group multiple Dirichlet series. III. Eisenstein series and twisted unstable $A_r$*, Ann. of Math. (2)**166**(2007), no. 1, 293–316. MR**2342698**, DOI 10.4007/annals.2007.166.293 - Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein,
*$p$-adic Whittaker functions on the metaplectic group*, Duke Math. J.**63**(1991), no. 2, 379–397. MR**1115113**, DOI 10.1215/S0012-7094-91-06316-7 - B. Brubaker. Analytic continuation for cubic multiple Dirichlet series. Thesis, Brown University, 2003.
- Gautam Chinta, Solomon Friedberg, and Paul E. Gunnells,
*On the $p$-parts of quadratic Weyl group multiple Dirichlet series*, J. Reine Angew. Math.**623**(2008), 1–23. MR**2458038**, DOI 10.1515/CRELLE.2008.070 - Gautam Chinta, Solomon Friedberg, and Jeffrey Hoffstein,
*Multiple Dirichlet series and automorphic forms*, Multiple Dirichlet series, automorphic forms, and analytic number theory, Proc. Sympos. Pure Math., vol. 75, Amer. Math. Soc., Providence, RI, 2006, pp. 3–41. MR**2279929**, DOI 10.1090/pspum/075/2279929 - G. Chinta and P. E. Gunnells. Weyl group multiple Dirichlet series of type ${A}_2$. Submitted to the Lang memorial volume.
- Gautam Chinta and Paul E. Gunnells,
*Weyl group multiple Dirichlet series constructed from quadratic characters*, Invent. Math.**167**(2007), no. 2, 327–353. MR**2270457**, DOI 10.1007/s00222-006-0014-1 - Gautam Chinta,
*Mean values of biquadratic zeta functions*, Invent. Math.**160**(2005), no. 1, 145–163. MR**2129710**, DOI 10.1007/s00222-004-0407-y - Gautam Chinta,
*Multiple Dirichlet series over rational function fields*, Acta Arith.**132**(2008), no. 4, 377–391. MR**2413360**, DOI 10.4064/aa132-4-7 - Adrian Diaconu, Dorian Goldfeld, and Jeffrey Hoffstein,
*Multiple Dirichlet series and moments of zeta and $L$-functions*, Compositio Math.**139**(2003), no. 3, 297–360. MR**2041614**, DOI 10.1023/B:COMP.0000018137.38458.68 - Benji Fisher and Solomon Friedberg,
*Sums of twisted $\rm GL(2)$ $L$-functions over function fields*, Duke Math. J.**117**(2003), no. 3, 543–570. MR**1979053**, DOI 10.1215/S0012-7094-03-11735-4 - Benji Fisher and Solomon Friedberg,
*Double Dirichlet series over function fields*, Compos. Math.**140**(2004), no. 3, 613–630. MR**2041772**, DOI 10.1112/S0010437X03000848 - Solomon Friedberg, Jeffrey Hoffstein, and Daniel Lieman,
*Double Dirichlet series and the $n$-th order twists of Hecke $L$-series*, Math. Ann.**327**(2003), no. 2, 315–338. MR**2015073**, DOI 10.1007/s00208-003-0455-4 - Dorian Goldfeld and Jeffrey Hoffstein,
*Eisenstein series of ${1\over 2}$-integral weight and the mean value of real Dirichlet $L$-series*, Invent. Math.**80**(1985), no. 2, 185–208. MR**788407**, DOI 10.1007/BF01388603 - Jeffrey Hoffstein,
*Theta functions on the $n$-fold metaplectic cover of $\textrm {SL}(2)$—the function field case*, Invent. Math.**107**(1992), no. 1, 61–86. MR**1135464**, DOI 10.1007/BF01231881 - James E. Humphreys,
*Reflection groups and Coxeter groups*, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR**1066460**, DOI 10.1017/CBO9780511623646 - Kenneth Ireland and Michael Rosen,
*A classical introduction to modern number theory*, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR**1070716**, DOI 10.1007/978-1-4757-2103-4 - D. A. Kazhdan and S. J. Patterson,
*Metaplectic forms*, Inst. Hautes Études Sci. Publ. Math.**59**(1984), 35–142. MR**743816**, DOI 10.1007/BF02698770 - Tomio Kubota,
*Some number-theoretical results on real analytic automorphic forms*, Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970) Lecture Notes in Math., Vol. 185, Springer, Berlin, 1971, pp. 87–96. MR**0314768** - Tomio Kubota,
*Some results concerning reciprocity law and real analytic automorphic functions*, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 382–395. MR**0340221** - Jürgen Neukirch,
*Algebraic number theory*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR**1697859**, DOI 10.1007/978-3-662-03983-0 - S. J. Patterson,
*A cubic analogue of the theta series*, J. Reine Angew. Math.**296**(1977), 125–161. MR**563068**, DOI 10.1515/crll.1977.296.125 - S. J. Patterson,
*A cubic analogue of the theta series. II*, J. Reine Angew. Math.**296**(1977), 217–220. MR**563069**, DOI 10.1515/crll.1977.296.217 - S. J. Patterson,
*Note on a paper of J. Hoffstein*, Glasg. Math. J.**49**(2007), no. 2, 243–255. MR**2347258**, DOI 10.1017/S0017089507003540

## Additional Information

**Gautam Chinta**- Affiliation: Department of Mathematics, The City College of CUNY, New York, New York 10031
- MR Author ID: 679536
- Email: chinta@sci.ccny.cuny.edu
**Paul E. Gunnells**- Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003
- Email: gunnells@math.umass.edu
- Received by editor(s): March 11, 2008
- Published electronically: July 31, 2009
- Additional Notes: Both authors thank the NSF for support.
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**23**(2010), 189-215 - MSC (2000): Primary 11F66, 11M41; Secondary 11F37, 11F70, 22E99
- DOI: https://doi.org/10.1090/S0894-0347-09-00641-9
- MathSciNet review: 2552251