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R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

Part of: Discontinuous groups and automorphic forms Lie groups

Published online by Cambridge University Press:  08 March 2021

Fan Gao Open the ORCID record for Fan Gao [Opens in a new window]
Fan Gao*
Affiliation:
School of Mathematical Sciences, Yuquan Campus, Zhejiang University, 38 Zheda Road, Hangzhou, China 310027 (gaofan@zju.edu.cn)
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Abstract

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For a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such a principal series representation. Moreover, for certain saturated covers of a semisimple simply connected group, we also propose a simpler conjectural formula for such dimensions. This latter conjectural formula is verified in several cases, including covers of the symplectic groups.

Keywords

covering groupsR-groupsWhittaker functionalsscattering matrixgamma factorPlancherel measure

MSC classification

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 1 , January 2023 , pp. 213 - 273
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Arthur, J., On some problems suggested by the trace formula, in Lie Group Representations, II (College Park, Md., 1982/1983), Lecture Notes in Mathematics, 1041, pp. 1–49 (Springer, Berlin, 1984).Google Scholar
Arthur, J., Unipotent automorphic representations: conjectures, II, Astérisque, 171–172 1989, pp. 13–71.Google Scholar
Arthur, J., On elliptic tempered characters, Acta Math. 171(1) (1993), 73–138.CrossRefGoogle Scholar
Arthur, J., A note on $L$ -packets, Pure Appl. Math. Q. 2(1), (2006), 199–217.CrossRefGoogle Scholar
Arthur, J., The endoscopic classification of representations, in Automorphic Representations and $L$ -Functions, Tata Institute of Fundamental Research Studies in Mathematics, 22, pp. 1–22 (Tata Institute of Fundamental Research, Mumbai, 2013).Google Scholar
Ban, D. and Goldberg, D., $R$ -groups and parameters, Pacific J. Math. 255(2) (2012), 281–303.CrossRefGoogle Scholar
Ban, D. and Zhang, Y., Arthur $R$ -groups, classical $R$ -groups, and Aubert involutions for $SO\left(2n+1\right)$ , Compos. Math. 141(2) (2005), 323–343.CrossRefGoogle Scholar
Barbasch, D. and Moy, A., Whittaker models with an Iwahori fixed vector, in Representation Theory and Analysis on Homogeneous Spaces, (New Brunswick, NJ, 1993), Contemporary Mathematics, 177, pp. 101–105 (American Mathematical Society, Providence, RI, 1994).CrossRefGoogle Scholar
Borel, A., Automorphic $L$ -functions, in Automorphic Forms, Representations and $L$ -Functions, (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Proceedings of Symposia in Pure Mathematics, XXXIII, pp. 27–61 (American Mathematical Society, Providence, RI, 1979).CrossRefGoogle Scholar
Bourbaki, N., Lie Groups and Lie Algebras (chapters 4–6), Elements of Mathematics (Berlin) (Springer-Verlag, Berlin, 2002). Translated from the 1968 French original by Pressley, A..CrossRefGoogle Scholar
Brylinski, J.-L. and Deligne, P., Central extensions of reductive groups by ${\boldsymbol{K}}_2$ , Publ. Math. Inst. Hautes Études Sci. 94 (2001), 5–85.CrossRefGoogle Scholar
Casselman, W. and Shalika, J., The unramified principal series of $p$ -adic groups. II. The Whittaker function, Compos. Math. 41(2) (1980), 207–231.Google Scholar
Chinta, G. and Offen, O., A metaplectic Casselman-Shalika formula for ${\text{GL}}_r$ , Amer. J. Math. 135(2) (2013), 403–441.CrossRefGoogle Scholar
Gan, W. T. and Gao, F., The Langlands-Weissman program for Brylinski-Deligne extensions, Astérisque 398 (2018), 187–275.Google Scholar
Gao, F., Distinguished theta representations for certain covering groups, Pacific J. Math. 290(2) (2017), 333–379.CrossRefGoogle Scholar
Gao, F., The Langlands-Shahidi L-functions for Brylinski-Deligne extensions, Amer. J. Math. 140(1) (2018), 83–137.CrossRefGoogle Scholar
Gao, F., Hecke $L$ -functions and Fourier coefficients of covering Eisenstein series, Preprint, 2018, https://sites.google.com/site/fangaonus/research.Google Scholar
Gao, F., Kazhdan–Lusztig representations and Whittaker space of some genuine representations, Math. Ann. 376(1) (2020), 289–358.CrossRefGoogle Scholar
Gao, F., Shahidi, F. and Szpruch, D., Local coefficients and gamma factors for principal series of covering groups, Mem. Amer. Math. Soc. (2019), https://arxiv.org/abs/1902.02686.Google Scholar
Gao, F. and Weissman, M. H., Whittaker models for depth zero representations of covering groups, Int. Math. Res. Not. IMRN 11 (2019), 3580–3620.CrossRefGoogle Scholar
Ginzburg, D., Non-generic unramified representations in metaplectic covering groups, Israel J. Math. 226(1) (2018), 447–474.CrossRefGoogle Scholar
Goldberg, D., Reducibility of induced representations for $\text{Sp}(2n)$ and $\text{SO}(n)$ , Amer. J. Math. 116(5) (1994), 1101–1151.CrossRefGoogle Scholar
Goldberg, D., On dual $R$ -groups for classical groups, in On Certain $L$ -Functions, Clay Mathematical Proceedings, 13, pp. 159–185 (American Mathematical Society, Providence, RI, 2011).Google Scholar
Kazhdan, D. A. and Patterson, S. J., Metaplectic forms, Publ. Math. Inst. Hautes Études Sci. 59 (1984), 35–142.CrossRefGoogle Scholar
Keys, C. D., On the decomposition of reducible principal series representations of $p$ -adic Chevalley groups, Pacific J. Math. 101(2) (1982), 351–388.CrossRefGoogle Scholar
Keys, C. D., Reducibility of unramified unitary principal series representations of $p$ -adic groups and class- $1$ representations, Math. Ann. 260(4) (1982), 397–402.CrossRefGoogle Scholar
Keys, C. D., $L$ -indistinguishability and $R$ -groups for quasisplit groups: unitary groups in even dimension, Ann. Sci. Éc. Norm. Supér. (4) 20(1) (1987), 31–64.CrossRefGoogle Scholar
Keys, C. D. and Shahidi, F., Artin $L$ -functions and normalization of intertwining operators, Ann. Sci. Éc. Norm. Supér. (4) 21(1) (1988), 67–89.CrossRefGoogle Scholar
Knapp, A. W. and Stein, E. M., Singular integrals and the principal series. IV, Proc. Natl. Acad. Sci. USA 72 (1975), 2459–2461.CrossRefGoogle ScholarPubMed
Labesse, J.-P. and Langlands, R. P., $L$ -indistinguishability for $\text{SL}(2)$ , Canad. J. Math. 31(4) (1979), 726–785.CrossRefGoogle Scholar
Langlands, R. P., On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics, 544 (Springer-Verlag, Berlin, 1976).CrossRefGoogle Scholar
Li, J.-S., Some results on the unramified principal series of $p$ -adic groups, Math. Ann. 292(4) (1992), 747–761.CrossRefGoogle Scholar
Li, W.-W., La formule des traces pour les revêtements de groupes réductifs connexes. II. Analyse harmonique locale, Ann. Sci. Éc. Norm. Supér. (4) 45(5) (2012), 787–859.CrossRefGoogle Scholar
Li, W.-W., La formule des traces pour les revêtements de groupes réductifs connexes. I. Le développement géométrique fin, J. Reine Angew. Math. 686 (2014), 37–109.Google Scholar
Luo, C., Knapp-Stein dimension theorem for finite central covering groups, Pacific J. Math. 306(1) (2020), 265–280.Google Scholar
McNamara, P. J., Principal series representations of metaplectic groups over local fields, in Multiple Dirichlet Series, L-Functions and Automorphic Forms, Progress in Mathematics, 300, pp. 299–327 (Birkhäuser/Springer, New York, 2012).CrossRefGoogle Scholar
McNamara, P. J., The metaplectic Casselman-Shalika formula, Trans. Amer. Math. Soc. 368(4) (2016), 2913–2937.CrossRefGoogle Scholar
Patterson, S. J., Metaplectic forms and Gauss sums. I, Compos. Math. 62(3) (1987), 343–366.Google Scholar
Shahidi, F., Some results on $L$ -indistinguishability for $\text{SL}(r)$ , Canad. J. Math. 35(6) (1983), 1075–1109.CrossRefGoogle Scholar
Shahidi, F., A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$ -adic groups, Ann. of Math. (2) 132(2) (1990), 273–330.Google Scholar
Shahidi, F., Arthur packets and the Ramanujan conjecture, Kyoto J. Math. 51(1) (2011), 1–23.CrossRefGoogle Scholar
Shelstad, D., Notes on $L$ -indistinguishability (based on a lecture of R. P. Langlands), in Automorphic Forms, Representations and $L$ -Functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Proceedings of Symposia in Pure Mathematics, XXXIII, pp. 193–203 (American Mathematical Society, Providence, RI, 1979).CrossRefGoogle Scholar
Shelstad, D., $L$ -indistinguishability for real groups, Math. Ann. 259(3) (1982), 385–430.Google Scholar
Silberger, A. J., The Knapp-Stein dimension theorem for $p$ -adic groups, Proc. Amer. Math. Soc. 68(2) (1978), 243–246.Google Scholar
Silberger, A. J., Correction: “The Knapp-Stein dimension theorem for $p$ -adic groups” [Proc. Amer. Math. Soc. 68 (1978), no. 2, 243–246; MR 58 #11245], Proc. Amer. Math. Soc. 76(1) (1979), 169–170.Google Scholar
Silberger, A. J., Introduction to Harmonic Analysis on Reductive $p$ -adic Groups, Mathematical Notes, 23 (Princeton University Press, Princeton, NJ, Tokyo, 1979). Based on lectures by Harish-Chandra at the Institute for Advanced Study, 1971–1973.Google Scholar
Steinberg, R., Lectures on Chevalley Groups, University Lecture Series, 66 (American Mathematical Society, Providence, RI, 2016).CrossRefGoogle Scholar
Szpruch, D., Symmetric genuine spherical Whittaker functions on $\overline{\text{GS}{p}_{2n}(F)}$ , Canad. J. Math. 67(1) (2015), 214–240.Google Scholar
Szpruch, D., On Shahidi local coefficients matrix, Manuscripta Math. 159(1–2) (2019), 117–159.CrossRefGoogle Scholar
Tate, J. T., Fourier Analysis in Number Fields, and Hecke’s Zeta-Runctions, in Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pp. 305–347 (Thompson, Washington, DC, 1967).Google Scholar
Weissman, M. H., Metaplectic tori over local fields, Pacific J. Math. 241(1) (2009), 169–200.CrossRefGoogle Scholar
Weissman, M. H., Split metaplectic groups and their L-groups, J. Reine Angew. Math. 696 (2014), 89–141.Google Scholar
Weissman, M. H., L-groups and parameters for covering groups, Astérisque 398 (2018), 33–186.Google Scholar
Winarsky, N., Reducibility of principal series representations of $p$ -adic Chevalley groups, Amer. J. Math. 100(5) (1978), 941–956.CrossRefGoogle Scholar