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Metaplectic Whittaker functions and crystal bases. (English) Zbl 1217.22015
The author studies Whittaker functions on nonlinear coverings of simple algebraic groups over a non-archimedean local field. He produces a recipe for expressing such a Whittaker function as a weighted sum over a crystal graph and shows that in type A, these expressions agree with known formulae for the prime-power-supported coefficients of multiple Dirichlet series.

MSC:
22E50Representations of Lie and linear algebraic groups over local fields
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References:
[1] J. E. Anderson, A polytope calculus for semisimple groups , Duke Math. J. 116 (2003), 567-588. · Zbl 1064.20047 · doi:10.1215/S0012-7094-03-11636-1
[2] P. Baumann and S. Gaussent, On Mirković-Vilonen cycles and crystal combinatorics , Represent. Theory 12 (2008), 83-130. · Zbl 1217.20028 · doi:10.1090/S1088-4165-08-00322-1
[3] A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties , Invent. Math. 143 (2001), 77-128. · Zbl 1061.17006 · doi:10.1007/s002220000102
[4] N. Bourbaki, Éléments de mathématique, f asc. 34: Groupes et algèbres de Lie, chapitres 4-6, Actualités Sci. Indust. 1337 , Hermann, Paris, 1968.
[5] A. Braverman, M. Finkelberg, and D. Gaitsgory, “Uhlenbeck spaces via affine Lie algebras” in The Unity of Mathematics , Progr. Math. 244 , Birkhäuser, Boston, 2006, 17-135. · Zbl 1105.14013 · doi:10.1007/0-8176-4467-9_2
[6] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmannian , Duke Math. J. 107 (2001), 561-575. · Zbl 1015.20030 · doi:10.1215/S0012-7094-01-10736-9
[7] 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 , Proc. Sympos. Pure Math. 75 , Amer. Math. Soc. Providence, 2006, 91-114. · Zbl 1112.11025
[8] B. Brubaker, D. Bump, and S. Friedberg, Weyl group multiple Dirichlet series, Eisenstein series and crystal bases , to appear in Ann. of Math. · Zbl 1187.11015
[9] F. Bruhat and J. Tits, Groupes réductifs sur un corps local , Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5-251. · Zbl 0254.14017 · doi:10.1007/BF02715544 · numdam:PMIHES_1972__41__5_0 · eudml:103918
[10] D. Bump and M. Nakasuji, Integration on $p$-adic groups and crystal bases , Proc. Amer. Math. Soc. 138 (2010), 1595-1605. · Zbl 1268.22009 · doi:10.1090/S0002-9939-09-10206-X
[11] W. Casselman and J. Shalika, The unramified principal series of $p$ -adic groups, II, The Whittaker function, Compositio Math. 41 (1980), 207-231. · Zbl 0472.22005 · numdam:CM_1980__41_2_207_0 · eudml:89456
[12] G. Chinta and P. E. Gunnells, Constructing Weyl group multiple Dirichlet series , J. Amer. Math. Soc. 23 (2010), 189-215. · Zbl 1254.11048 · doi:10.1090/S0894-0347-09-00641-9
[13] G. Chinta and O. Offen, A metaplectic Casselman-Shalika formula for ${\rm GL}_r$ , Ann. of Math. (2) 171 (2010), 245-294.
[14] J. Kamnitzer, Mirkovic-Vilonen cycles and polytopes , to appear in Ann. of Math. · Zbl 1271.20058
[15] M. Kashiwara, “On crystal bases” in Representations of Groups (Banff, Alberta , 1994), CMS Conf. Proc. 16 , Amer. Math. Soc. Providence, 1995, 155-197. · Zbl 0851.17014
[16] 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 · numdam:PMIHES_1984__59__35_0 · eudml:103999
[17] G. Lusztig, Canonical bases arising from quantized enveloping algebras , J. Amer. Math. Soc. 3 (1990), 447-498. JSTOR: · Zbl 0703.17008 · doi:10.2307/1990961 · http://links.jstor.org/sici?sici=0894-0347%28199004%293%3A2%3C447%3ACBAFQE%3E2.0.CO%3B2-F&origin=euclid
[18] -, “Introduction to quantized enveloping algebras” in New developments in Lie Theory and Their Applications (Córdoba, Spain, 1989) , Progr. Math. 105 , Birkhäuser, Boston, 1992, 49-65. · Zbl 0767.17014
[19] -, Introduction to Quantum Groups , Progr. Math. 110 , Birkhäuser, Boston, 1993. · Zbl 0788.17010
[20] -, An algebraic-geometric parametrization of the canonical basis , Adv. Math. 120 (1996), 173-190. · Zbl 0877.17006 · doi:10.1006/aima.1996.0036
[21] H. Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés , Ann. Sci. École Norm. Sup. 2 (1969), 1-62. · Zbl 0261.20025 · numdam:ASENS_1969_4_2_1_1_0 · eudml:81843
[22] P. Mcnamara, Principal series representations of metaplectic groups over local fields , · Zbl 0547.22009 · numdam:CM_1984__51_1_115_0 · eudml:89629
[23] I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings , Ann. of Math. (2) 166 (2007), 95-143. · Zbl 1138.22013 · doi:10.4007/annals.2007.166.95
[24] C. Moore, Group extensions of $p$ -adic and adelic linear groups, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 157-222. · Zbl 0159.03203 · doi:10.1007/BF02698923 · numdam:PMIHES_1968__35__5_0 · eudml:103886
[25] G. Savin, On unramified representations of covering groups , J. Reine Angew. Math. 566 (2004), 111-134. · Zbl 1032.22006 · doi:10.1515/crll.2004.001
[26] R. Steinberg, Lectures on Chevalley groups , notes prepared by John Faulkner and Robert Wilson, Yale University, New Haven, 1968. · Zbl 1196.22001