Abstract
We prove a formula for the dimension of Whittaker functionals of irreducible constituents of a regular unramified genuine principal series for covering groups. The formula explicitly relates such dimension to the Kazhdan–Lusztig representations associated with certain right cells of the Weyl group. We also state a refined version of the formula, which is proved under some natural assumption. The refined formula is also verified unconditionally in several important cases.
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References
Aluffi, P., Mihalcea, L.C., Schürmann, J., Su, C.: Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman’s problem. ar**v:1902.10101v1
Ban, D., Jantzen, C.: The Langlands quotient theorem for finite central extensions of \(p\)-adic groups. Glas. Mat. Ser. III 48(68)(2), 313–334 (2013). https://doi.org/10.3336/gm.48.2.07
Ban, D., Jantzen, C.: The Langlands quotient theorem for finite central extensions of \(p\)-adic groups II: intertwining operators and duality. Glas. Mat. Ser. III 51(71)(1), 153–163 (2016)
Banks, W., Bump, D., Lieman, D.: Whittaker–Fourier coefficients of metaplectic Eisenstein series. Compos. Math. 135(2), 153–178 (2003). https://doi.org/10.1023/A:1021763918640
Barbasch, D., Vogan, D.: Primitive ideals and orbital integrals in complex classical groups. Math. Ann. 259(2), 153–199 (1982)
Barbasch, D., Vogan, D.: Primitive ideals and orbital integrals in complex exceptional groups. J. Algebra 80(2), 350–382 (1983)
Beĭ linson, A., Bernstein, J.: Localisation de \(g\)-modules. C. R. Acad. Sci. Paris Sér. I Math. 292(1), 15–18 (1981). (French, with English summary)
Benson, C.T., Curtis, C.W.: On the degrees and rationality of certain characters of finite Chevalley groups. Trans. Am. Math. Soc. 165, 251–273 (1972)
Bernšteĭn, I.N., Gel’fand, I.M., Gel’fand, S.I.: Schubert cells, and the cohomology of the spaces \(G/P\). Uspehi Mat. Nauk. 28(3(171)), 3–26 (1973). (Russian)
Bernstein, I.N., Zelevinsky, A.V.: Induced representations of reductive \(\mathfrak{p}\)-adic groups. I. Ann. Sci. École Norm. Sup. (4) 10(4), 441–472 (1977)
Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics. Springer, New York (2005)
Blondel, C.: Uniqueness of Whittaker model for some supercuspidal representations of the metaplectic group. Compos. Math. 83(1), 1–18 (1992)
Blondel, C., Stevens, S.: Genericity of supercuspidal representations of \(p\)-adic \(\text{ Sp }_4\). Compos. Math. 145(1), 213–246 (2009). https://doi.org/10.1112/S0010437X08003849
Borel, A.: Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup. Invent. Math. 35, 233–259 (1976). https://doi.org/10.1007/BF01390139
Borel, A.: Automorphic \(L\)-functions, automorphic forms, representations and \(L\)-functions. In: Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977, Proc. Sympos. Pure Math., XXXIII, pp. 27–61. Amer. Math. Soc., Providence, RI (1979)
Bourbaki, N.: Lie Groups and Lie Algebras, Chapters 4–6. Elements of Mathematics. Springer, Berlin (2002). (Translated from the 1968 French original by Andrew Pressley)
Brylinski, J.-L., Deligne, P.: Central extensions of reductive groups by \( {K}_2\). Publ. Math. Inst. Hautes Études Sci. 94, 5–85 (2001). https://doi.org/10.1007/s10240-001-8192-2
Brylinski, J.-L., Kashiwara, M.: Kazhdan–Lusztig conjecture and holonomic systems. Invent. Math. 64(3), 387–410 (1981)
Bump, D., Nakasuji, M.: Casselman’s basis of Iwahori vectors and the Bruhat order. Can. J. Math. 63(6), 1238–1253 (2011)
Bump, D., Nakasuji, M.: Casselman’s basis of Iwahori vectors and Kazhdan–Lusztig polynomials. ar**v:1710.03185
Cai, Y.: Fourier coefficients for theta representations on covers of general linear groups. Trans. Am. Math. Soc. ar**v:1602.06614
Cai, Y., Friedberg, S., Ginzburg, D., Kaplan, E.: Doubling constructions and tensor product L-functions: the linear case. Invent. Math. ar**v:1710.00905
Cai, Y., Friedberg, S., Kaplan, E.: Doubling constructions: local and global theory, with an application to global functoriality for non-generic cuspidal representations. ar**v:1802.02637
Carter, R.W.: Finite Groups of Lie Type. Conjugacy Classes and Complex Characters. Wiley, Chichester (1993)
Casselman, W.: Introduction to the theory of admissible representations of \(p\)-adic reductive groups. https://www.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf
Casselman, W., Shahidi, F.: On irreducibility of standard modules for generic representations. Ann. Sci. École Norm. Sup. (4) 31(4), 561–589 (1998). https://doi.org/10.1016/S0012-9593(98)80107-9. (English, with English and French summaries)
Deodhar, V.: A brief survey of Kazhdan–Lusztig theory and related topics. Algebraic groups and their generalizations: classical methods, (University Park, PA, 1991). In: Proc. Sympos. Pure Math., vol. 56, pp. 105–124. Amer. Math. Soc., Providence, RI (1994)
Gan, W.T., Gao, F.: The Langlands–Weissman program for Brylinski–Deligne extensions. Astérisque 398, 187–275 (2018). (English, with English and French summaries)
Gan, W.T., Gao, F., Weissman, M.H.: L-group and the Langlands program for covering groups: a historical introduction. Astérisque 398, 1–31 (2018). (English, with English and French summaries)
Gao, F.: Distinguished theta representations for certain covering groups. Pac. J. Math. 290(2), 333–379 (2017). https://doi.org/10.2140/pjm.2017.290.333
Gao, F.: The Langlands–Shahidi L-functions for Brylinski–Deligne extensions. Am. J. Math. 140(1), 83–137 (2018). https://doi.org/10.1353/ajm.2018.0001
Gao, F.: Hecke \(L\)-functions and Fourier coefficients of covering Eisenstein series. https://sites.google.com/site/fangaonus/research
Gao, F.: R-group and Whittaker space of some genuine representations. (preprint)
Gao, F., Shahidi, F., Szpruch, D.: On the local coefficients matrix for coverings of \(\text{ SL }_2\). Geometry, algebra, number theory, and their information technology applications. In: Springer Proc. Math. Stat., vol. 251, pp. 207–244. Springer, Cham (2018)
Gao, F., Shahidi, F., Szpruch, D.: Gamma factor for genuine principal series of covering groups (with an appendix by Caihua Luo). ar**v:1902.02686
Gao, F., Weissman, M.H.: Whittaker models for depth zero representations of covering groups. Int. Math. Res. Not. IMRN 11, 3580–3620 (2019). https://doi.org/10.1093/imrn/rnx235
Gelfand, I.M., Kazhdan, D.A.: Representations of the group \(\text{ GL }(n,K)\) where \(K\) is a local field. Lie groups and their representations. In: Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), pp. 95–118. Halsted, New York (1975)
Ginzburg, D.: Non-generic unramified representations in metaplectic covering groups. Isr. J. Math. 226(1), 447–474 (2018). https://doi.org/10.1007/s11856-018-1702-4
Heiermann, V., Muić, G.: On the standard modules conjecture. Math. Z. 255(4), 847–853 (2007). https://doi.org/10.1007/s00209-006-0052-9
Heiermann, V., Opdam, E.: On the tempered \(L\)-functions conjecture. Am. J. Math. 135(3), 777–799 (2013). https://doi.org/10.1353/ajm.2013.0026
Hiller, H.: Geometry of Coxeter groups. Research Notes in Mathematics, vol. 54. Pitman (Advanced Publishing Program), Boston (1982)
Howe, R., Piatetski-Shapiro, I.I.: A counterexample to the “generalized Ramanujan conjecture” for (quasi-) split groups, Automorphic forms, representations and \(L\)-functions. In: Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Proc. Sympos. Pure Math., XXXIII, pp. 315–322. Amer. Math. Soc., Providence, RI (1979)
Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)
Joseph, A.: Goldie rank in the envelo** algebra of a semisimple Lie algebra, I. J. Algebra 65(2), 269–283 (1980)
Kaplan, E.: Doubling constructions and tensor product L-functions: coverings of the symplectic group. ar**v:1902.00880
Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53(2), 165–184 (1979)
Kazhdan, D., Lusztig, G.: Proof of the Deligne–Langlands conjecture for Hecke algebras. Invent. Math. 87(1), 153–215 (1987)
Kazhdan, D.A., Patterson, S.J.: Metaplectic forms. Inst. Hautes Études Sci. Publ. Math. 59, 35–142 (1984)
Leslie, S.: A generalized theta lifting, CAP representations, and Arthur parameters. ar**v:1703.02597
Lusztig, G.: On a theorem of Benson and Curtis. J. Algebra 71(2), 490–498 (1981)
Lusztig, G.: Some examples of square integrable representations of semisimple \(p\)-adic groups. Trans. Am. Math. Soc. 277(2), 623–653 (1983)
Lusztig, G.: Left Cells in Weyl Groups. Lie Group Representations, I. Lecture Notes in Math, vol. 1024, pp. 99–111. Springer, Berlin (1983)
Lusztig, G.: Characters of Reductive Groups Over a Finite Field. Annals of Mathematics Studies, vol. 107. Princeton University Press, Princeton (1984)
Lusztig, G.: Hecke Algebras with Unequal Parameters. CRM Monograph Series, vol. 18. American Mathematical Society, Providence (2003)
McNamara, P.J.: Principal series representations of metaplectic groups over local fields. Multiple Dirichlet series, L-functions and automorphic forms. In: Progr. Math., vol. 300, pp. 299–327. Birkhäuser/Springer, New York (2012). https://doi.org/10.1007/978-0-8176-8334-413
McNamara, P.J.: The metaplectic Casselman–Shalika formula. Trans. Am. Math. Soc. 368(4), 2913–2937 (2016). https://doi.org/10.1090/tran/6597
Mœglin, C., Waldspurger, J.-L.: Modèles de Whittaker dégénérés pour des groupes \(p\)-adiques. Math. Z. 196(3), 427–452 (1987). (French)
Reeder, M.: On certain Iwahori invariants in the unramified principal series. Pac. J. Math. 153(2), 313–342 (1992)
Robinson, G.B.: On the representations of the symmetric group. Am. J. Math. 60(3), 745–760 (1938)
Robinson, G.B.: On the representations of the symmetric group. II. Am. J. Math. 69, 286–298 (1947)
Robinson, G.B.: On the representations of the symmetric group III. Am. J. Math. 70, 277–294 (1948)
Rodier, F.: Whittaker models for admissible representations of reductive \(p\)-adic split groups. Harmonic analysis on homogeneous spaces. In: Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972), pp. 425–430. Amer. Math. Soc., Providence, RI (1973)
Rodier, F.: Décomposition de la série principale des groupes réductifs \(p\)-adiques. Noncommutative harmonic analysis and Lie groups, (Marseille, 1980), Lecture Notes in Math., vol. 880, pp. 408–424. Springer, Berlin-New York (1981) (French)
Rodier, F.: Decomposition of principal series for reductive \(p\)-adic groups and the Langlands’ classification. Operator algebras and group representations, Vol. II, (Neptun, 1980), Monogr. Stud. Math., vol. 18, pp. 86–94. Pitman, Boston, MA (1984)
Rogawski, J.D.: On modules over the Hecke algebra of a \(p\)-adic group. Invent. Math. 79(3), 443–465 (1985)
Rohrlich, D.E.: Elliptic curves and the Weil–Deligne group. Elliptic curves and related topics. In: CRM Proc. Lecture Notes, vol. 4, pp. 125–157. Amer. Math. Soc., Providence, RI (1994)
Roichman, Y.: Induction and restriction of Kazhdan–Lusztig cells. Adv. Math. 134(2), 384–398 (1998)
Savin, G.: On unramified representations of covering groups. J. Reine Angew. Math. 566, 111–134 (2004)
Schensted, C.: Longest increasing and decreasing subsequences. Can. J. Math. 13, 179–191 (1961)
Shalika, J.A.: The multiplicity one theorem for \(\text{ GL }_{n}\). Ann. Math. (2) 100, 171–193 (1974). https://doi.org/10.2307/1971071
Shi, J.Y.: The Kazhdan–Lusztig Cells in Certain Affine Weyl Groups. Lecture Notes in Mathematics, vol. 1179. Springer, Berlin (1986)
Srinivasan, B.: The characters of the finite symplectic group \(\text{ Sp }(4,\, q)\). Trans. Am. Math. Soc. 131, 488–525 (1968). https://doi.org/10.2307/1994960
Steinberg, R.: Lectures on Chevalley groups, University Lecture Series, vol. 66. American Mathematical Society, Providence, RI (2016)
Suzuki, T.: Metaplectic Eisenstein series and the Bump–Hoffstein conjecture. Duke Math. J. 90(3), 577–630 (1997). https://doi.org/10.1215/S0012-7094-97-09016-5
Suzuki, T.: Distinguished representations of metaplectic groups. Am. J. Math. 120(4), 723–755 (1998)
Tate, J.T.: Number theoretic background. Automorphic forms, representations and \(L\)-functions. In: Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Proc. Sympos. Pure Math., XXXIII, pp. 3–26. Amer. Math. Soc., Providence, RI (1979)
Weissman, M.H.: Metaplectic tori over local fields. Pac. J. Math. 241(1), 169–200 (2009). https://doi.org/10.2140/pjm.2009.241.169
Weissman, M.H.: Split metaplectic groups and their L-groups. J. Reine Angew. Math. 696, 89–141 (2014). https://doi.org/10.1515/crelle-2012-0111
Weissman, M.H.: L-groups and parameters for covering groups. Astérisque 398, 33–186 (2018). (English, with English and French summaries)
Acknowledgements
I would like to thank Caihua Luo for several discussions on the content of Sect. 3. Thanks are also due to the referee for his or her careful reading and insightful comments.
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Communicated by Wei Zhang.
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Gao, F. Kazhdan–Lusztig representations and Whittaker space of some genuine representations. Math. Ann. 376, 289–358 (2020). https://doi.org/10.1007/s00208-019-01925-1
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DOI: https://doi.org/10.1007/s00208-019-01925-1