Weyl group multiple Dirichlet series, Eisenstein series and crystal bases. (English) Zbl 1302.11032

This important paper establishes part of the Eisenstein conjecture. Specifically it is proved that the Whittaker coefficients of Borel Eisenstein series on the metaplectic covers of \(\mathrm{GL}_{r+1}\) can be expressed in terms of Weyl group multiple Dirichlet series in \(r\) complex variables. To construct these Weyl group multiple Dirichlet series, the authors first attach quantities of number-theoretic nature to the vertices of a crystal graph. The “\(p\)-parts” of the coefficients of the Weyl group multiple Dirichlet series are then derived from the sum over those vertices.
In contrast to the non-metaplectic case, the coefficients of the Weyl group multiple Dirichlet series considered in the paper are not multiplicative. However, the authors prove that they satisfy a “twisted multiplicativity” property which suffices for determining the coefficients from their “\(p\)-parts”. The expression for the \(p\)-parts of these coefficients established in the paper can be considered as an extension of the formula of W. Casselman and J. Shalika [Compos. Math. 41, 207–231 (1980; Zbl 0472.22005)] and is thus of independent interest.
The authors show that the crystal graph description they give is equivalent to the Geldfand-Tsetlin description previously conjectured by the authors and J. Hoffstein [Ann. Math. (2) 166, No. 1, 293–316 (2007; Zbl 1154.11016)]. The \(p\)-parts of the coefficients were also described by G. Chinta and P. E. Gunnells [Invent. Math. 167, No. 2, 327–353 (2007; Zbl 1203.11062)] using averaging. Subsequent work by G. Chinta and O. Offen [Am. J. Math. 135, No. 2, 403–441 (2013; Zbl 1294.22012)] and by P. J. McNamara [Duke Math. J. 156, No. 1, 1–31 (2011; Zbl 1217.22015)] implies that the crystal graph description given in the paper under review agrees, in the case of type A, with the description given by Chinta and Gunnells [loc. cit.].


11F68 Dirichlet series in several complex variables associated to automorphic forms; Weyl group multiple Dirichlet series
17B37 Quantum groups (quantized enveloping algebras) and related deformations
Full Text: DOI


[1] W. D. Banks, J. Levy, and M. R. Sepanski, ”Block-compatible metaplectic cocycles,” J. Reine Angew. Math., vol. 507, pp. 131-163, 1999. · Zbl 0914.20039 · doi:10.1515/crll.1999.507.131
[2] J. Beineke, B. Brubaker, and S. Frechette, Weyl group multiple Dirichlet series of Type C, preprint, 2010. · Zbl 1277.11049 · doi:10.2140/pjm.2011.254.11
[3] A. Berenstein and A. Zelevinsky, ”Canonical bases for the quantum group of type \(A_r\) and piecewise-linear combinatorics,” Duke Math. J., vol. 82, iss. 3, pp. 473-502, 1996. · Zbl 0898.17006 · doi:10.1215/S0012-7094-96-08221-6
[4] A. Berenstein and A. Zelevinsky, ”Tensor product multiplicities, canonical bases and totally positive varieties,” Invent. Math., vol. 143, iss. 1, pp. 77-128, 2001. · Zbl 1061.17006 · doi:10.1007/s002220000102
[5] B. Brubaker and D. Bump, ”On Kubota’s Dirichlet series,” J. Reine Angew. Math., vol. 598, pp. 159-184, 2006. · Zbl 1113.11028 · doi:10.1515/CRELLE.2006.073
[6] B. Brubaker, D. Bump, G. Chinta, S. Friedberg, and J. Hoffstein, ”Weyl group multiple Dirichlet series. I,” in Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2006, vol. 75, pp. 91-114. · Zbl 1112.11025
[7] B. Brubaker, D. Bump, G. Chinta, and P. Gunnells, Crystals of Type B and metaplectic Whittaker functions (preprint). · Zbl 1370.11061
[8] B. Brubaker, D. Bump, and S. Friedberg, ”Weyl group multiple Dirichlet series. II. The stable case,” Invent. Math., vol. 165, iss. 2, pp. 325-355, 2006. · Zbl 1138.11017 · doi:10.1007/s00222-005-0496-2
[9] B. Brubaker, D. Bump, and S. Friedberg, Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory, Princeton, NJ: Princeton Univ. Press, 2011, vol. 175. · Zbl 1288.11052 · doi:10.1515/9781400838998
[10] B. Brubaker, D. Bump, and S. Friedberg, ”Twisted Weyl group multiple Dirichlet series: the stable case,” in Eisenstein Series and Applications, Boston, MA: Birkhäuser, 2008, vol. 258, pp. 1-26. · Zbl 1246.11107 · doi:10.1007/978-0-8176-4639-4_1
[11] B. Brubaker, D. Bump, and S. Friedberg, ”Gauss sum combinatorics and metaplectic Eisenstein series,” in Automorphic Forms and \(L\)-functions I. Global Aspects, Providence, RI: Amer. Math. Soc., 2009, vol. 488, pp. 61-81. · Zbl 1187.11015
[12] 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., vol. 166, iss. 1, pp. 293-316, 2007. · Zbl 1154.11016 · doi:10.4007/annals.2007.166.293
[13] W. Casselman and J. Shalika, ”The unramified principal series of \(p\)-adic groups. II. The Whittaker function,” Compositio Math., vol. 41, iss. 2, pp. 207-231, 1980. · Zbl 0472.22005
[14] G. Chinta, S. Friedberg, and P. E. Gunnells, ”On the \(p\)-parts of quadratic Weyl group multiple Dirichlet series,” J. Reine Angew. Math., vol. 623, pp. 1-23, 2008. · Zbl 1162.11026 · doi:10.1515/CRELLE.2008.070
[15] G. Chinta and P. E. Gunnells, ”Weyl group multiple Dirichlet series constructed from quadratic characters,” Invent. Math., vol. 167, iss. 2, pp. 327-353, 2007. · Zbl 1203.11062 · doi:10.1007/s00222-006-0014-1
[16] G. Chinta and P. E. Gunnells, ”Constructing Weyl group multiple Dirichlet series,” J. Amer. Math. Soc., vol. 23, iss. 1, pp. 189-215, 2010. · Zbl 1254.11048 · doi:10.1090/S0894-0347-09-00641-9
[17] G. Chinta and O. Offen, A metaplectic Casselman-Shalika formula for \({{ GL}}_r\), 2009. · Zbl 1294.22012
[18] I. M. Gel\('\)fand and D. A. Kajdan, ”Representations of the group \({ GL}(n,K)\) where \(K\) is a local field,” in Lie Groups and their Representations, New York: Halsted, 1975, pp. 95-118. · Zbl 0348.22011
[19] H. Jacquet, ”Fonctions de Whittaker associées aux groupes de Chevalley,” Bull. Soc. Math. France, vol. 95, pp. 243-309, 1967. · Zbl 0155.05901
[20] M. Kashiwara, ”Crystalizing the \(q\)-analogue of universal enveloping algebras,” Comm. Math. Phys., vol. 133, iss. 2, pp. 249-260, 1990. · Zbl 0724.17009 · doi:10.1007/BF02097367
[21] D. A. Kazhdan and S. J. Patterson, ”Metaplectic forms,” Inst. Hautes Études Sci. Publ. Math., iss. 59, pp. 35-142, 1984. · Zbl 0559.10026 · doi:10.1007/BF02698770
[22] A. N. Kirillov and A. D. Berenstein, ”Groups generated by involutions, Gel\('\)fand-Tsetlin patterns, and combinatorics of Young tableaux,” Algebra i Analiz, vol. 7, iss. 1, pp. 92-152, 1995. · Zbl 0848.20007
[23] T. Kubota, On Automorphic Functions and the Reciprocity Law in a Number Field, Tokyo: Kinokuniya Book-Store Co. Ltd., 1969, vol. 2. · Zbl 0231.10017
[24] C. Lenart, ”On the combinatorics of crystal graphs. I. Lusztig’s involution,” Adv. Math., vol. 211, iss. 1, pp. 204-243, 2007. · Zbl 1129.05058 · doi:10.1016/j.aim.2006.08.002
[25] P. Littelmann, ”Cones, crystals, and patterns,” Transform. Groups, vol. 3, iss. 2, pp. 145-179, 1998. · Zbl 0908.17010 · doi:10.1007/BF01236431
[26] G. Lusztig, Introduction to Quantum Groups, Boston, MA: Birkhäuser, 1993, vol. 110. · Zbl 0788.17010
[27] H. Matsumoto, ”Sur les sous-groupes arithmétiques des groupes semi-simples déployés,” Ann. Sci. École Norm. Sup., vol. 2, pp. 1-62, 1969. · Zbl 0261.20025
[28] P. McNamara, ”Metaplectic Whittaker functions and crystal bases,” Duke Math. J., vol. 156, pp. 1-31, 2011. · Zbl 1217.22015 · doi:10.1215/00127094-2010-064
[29] C. Moeglin and J. L. Waldspurger, Spectral Decomposition and Eisenstein Series, Une Paraphrase de l’Écriture, Cambridge: , 1995, vol. 113. · Zbl 1141.11029
[30] C. C. Moore, ”Group extensions of \(p\)-adic and adelic linear groups,” Inst. Hautes Études Sci. Publ. Math., iss. 35, pp. 5-70, 1968. · Zbl 1141.11029
[31] J. Neukirch, Algebraic Number Theory, New York: Springer-Verlag, 1999, vol. 322. · Zbl 0956.11021
[32] I. I. Pjateckij-vSapiro, ”Euler subgroups,” in Lie Groups and Their Representations, Proc. Summer School, Bolyai János Math. Soc., 1975, pp. 597-620. · Zbl 0329.20028
[33] M. -P. Schützenberger, ”La correspondance de Robinson,” in Combinatoire et Représentation du Groupe Symétrique, New York: Springer-Verlag, 1977, vol. 579, pp. 59-113. · Zbl 0398.05011
[34] J. A. Shalika, ”The multiplicity one theorem for \({ GL}_n\),” Ann. of Math., vol. 100, pp. 171-193, 1974. · Zbl 0316.12010 · doi:10.2307/1971071
[35] T. Shintani, ”On an explicit formula for class-\(1\) “Whittaker functions” on \(GL_n\) over \(P\)-adic fields,” Proc. Japan Acad., vol. 52, iss. 4, pp. 180-182, 1976. · Zbl 0387.43002 · doi:10.3792/pja/1195518347
[36] T. Tokuyama, ”A generating function of strict Gel\('\)fand patterns and some formulas on characters of general linear groups,” J. Math. Soc. Japan, vol. 40, iss. 4, pp. 671-685, 1988. · Zbl 0639.20022 · doi:10.2969/jmsj/04040671
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.