Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-17T02:14:10.139Z Has data issue: false hasContentIssue false
  • English
  • Français

Branching Rules for n-fold Covering Groups of SL2 over a Non-Archimedean Local Field

Published online by Cambridge University Press:  20 November 2018

Camelia Karimianpour
Camelia Karimianpour*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI., USA, e-mail : [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $G$ be the $n$-fold covering group of the special linear group of degree two over a non-Archimedean local field. We determine the decomposition into irreducibles of the restriction of the principal series representations of $G$ to a maximal compact subgroup. Moreover, we analyse those features that distinguish this decomposition from the linear case.

Keywords

20G05local fieldcovering grouprepresentationHilbert symbolK-type
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 3 , 01 September 2018 , pp. 553 - 571
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Aritürk, H., On the composition series ofprincipal series representations of a three-fold covering group of SL(2, K). NagoyaMath. J. 77(1980), 177–196. http://dx.doi.Org/10.1017/S0027763000018754Google Scholar
[2] Breuil, C. and Mezard, A., Multiplicites modulaires et representations de GL2(Zp) et de Gal(Qp/Qp) enl = p. Duke Math. J. 115 (2002), no. 2, 205–310. http://dx.doi.org/10.1215/S0012-7094-02-11522-1Google Scholar
[3] Bushnell, C. J. and Kutzko, P. C., Smooth representations of reductive p-adic groups: structure theory via types. Proc. London Math. Soc. (3) 77 (1998), no. 3, 582–634. http://dx.doi.Org/10.1112/S0024611598000574Google Scholar
[4] Campbell, P. S. and Nevins, M., Branching rulesfor unramified principal series representations of GL(3) over a p-adicfield. J. Algebra 321 (2009), no. 9, 2422–2444. http://dx.doi.Org/10.1016/j.jalgebra.2009.01.013Google Scholar
[5] Campbell, P. S. and Nevins, M., Branching rules for ramified principal series representations ofGL(3) over a p-adic field. Canad. J. Math. 62(2010), 34–51. http://dx.doi.Org/10.4153/CJM-2O10-003-5Google Scholar
[6] Casselman, W., The restriction ofa representation ofGL2(k) to GL2(o). Math. Ann. 206(1973), 311–318. http://dx.doi.Org/10.1007/BF01355984Google Scholar
[7] Digne, F. and Michel, J., Representations offinite groups of Lie type. London Mathematical Society Student Texts, 21, Cambridge University Press, Cambridge, 1991. http://dx.doi.Org/10.1017/CBO9781139172417Google Scholar
[8] GeFfand, I. M., Graev, M. I., and Pyatetskii-Shapiro, I. I., Representation theory and automorphic functions. Generalized Functions, 6, Academic Press, Inc., Boston, MA, 1990.Google Scholar
[9] Goldberg, D. and Szpruch, D., Plancherel measures for coverings of p-adic SL2(F). Int. J. Number Theory 12(2016), 1907–1936. http://dx.doi.Org/10.1142/S1 793042116501189Google Scholar
[10] Howard, T. K. and Weissman, M. H., Depth-zero representations of nonlinear Covers of p-adic groups. Int. Math. Res. Not. IMRN 2009, no. 21, 3979-3995. http://dx.doi.Org/10.1093/imrn/rnp076Google Scholar
[11] Joyner, D., A correspondence for the generalized Hecke algebra ofthe metaplectic cover SL(2,F), F p-adic. New York J. Math. 4(1998), 223–235.Google Scholar
[12] Joyner, D., Invariant distributions on the n-fold metaplectic Covers of GL(r, F), F p-adic. J. Fourier Anal. Appl. 7 (2001), no. 4, 343–358. http://dx.doi.org/10.1007/BF02514501Google Scholar
[13] Karimianpour, C., The Stone-von Neumann construction in branching rules and minimal degree Problems. PhD Thesis, University of Ottawa, 2015.Google Scholar
[14] Kazhdan, D. A. and Patterson, S. J., Metaplectic forms. Inst. Hautes Etudes Sei. Publ. Math. 59(1984), 35–142.Google Scholar
[15] Kubota, T., Topological covering of SL(2) over a localfield. J. Math. Soc. Japan 19(1967), 114–121. http://dx.doi.org/!0.2969/jmsj/01910114Google Scholar
[16] Kubota, T., On automorphic functions and the reciprocity law in a numberfield. Lectures in Mathematics, Department of Mathematics, Kyoto University, no. 2, Kinokuniya Book-Store Co., Ltd., Tokyo, 1969.Google Scholar
[17] Latham, P., Unicity oftypesfor supercuspidal representations of p-adic SL2. J. Number Theory 162(2016), 376–390. http://dx.doi.Org/10.1016/j.jnt.2015.10.008Google Scholar
[18] Matsumoto, H., Sur les sous-groupes arithmetiques des groupes semi-simples deployes. Ann. Sei. Ecole Norm. Sup. (4) 2(1969), 1–62. http://dx.doi.Org/10.24033/asens.11 74Google Scholar
[19] McNamara, P. J., Principal series representations of metaplectic groups over localfields. In: Multiple Dirichlet series, L-functions and automorphic forms, Progr. Math., 300, Birkhäuser/Springer, New York, 2012, pp. 299–327. http://dx.doi.org/10.1007/978-0-8176-8334-4J3Google Scholar
[20] Moen, C., Irreducibility of unitary prineipal series for covering groups of SL(2, k). Pacific J. Math. 135(1988), 89–110. http://dx.doi.org/10.2140/pjm.1988.135.89Google Scholar
[21] Moore, C. C., Group extensions ofp-adic and adelic linear groups. Inst. Hautes Etudes Sei. Publ. Math. 1968, no. 35, 157–222.Google Scholar
[22] Nadimpalli, S., Typical representations for level zero Bernstein components of GL., (F). J. Algebra 469(2017), 1–29. http://dx.doi.Org/10.1016/j.jalgebra.2016.08.033Google Scholar
[23] Nevins, M., Branching rules for prineipal series representations of SL(2) over a p-adic field. Canad. J. Math. 57 (2005), no. 3, 648–672. http://dx.doi.org/10.4153/CJM-2005-026-1Google Scholar
[24] Nevins, M., Patterns in branching rulesfor irreducible representations of SL2 (k),for k a p-adic field. In: Harmonie analysis on reduetive, p-adic groups, Contemp. Math., 543, American Mathematical Society, Providence, RI, 2011, pp. 185-199. http://dx.doi.Org/10.1090/conm/543/10735Google Scholar
[25] Nevins, M., Branching rulesfor supercuspidal representations of SL2 (k), for k a p-adic field. J. Algebra 377(2013), 204–231. http://dx.doi.Org/10.1016/j.jalgebra.2O12.12.003Google Scholar
[26] Onn, U. and Singla, P., On the unramifiedprineipal series of GL(3) over non-Archimedean local fields. J. Algebra 397(2014), 1–17. http://dx.doi.Org/1 0.101 6/j.jalgebra.2O13.08.022Google Scholar
[27] Paskunas, V., Unicity oftypesfor supercuspidal representations of GL2. Proc. London Math. Soc. (3) 91 (2005), no. 3, 623–654. http://dx.doi.Org/10.1112/S0024611505015340Google Scholar
[28] Savin, G., On unramified representations of covering groups. J. Reine Angew. Math. 566(2004), 111–134. http://dx.doi.Org/1 0.1 51 5/crll.2004.001Google Scholar
[29] Serre, J.-P., Localfields. Graduate Texts in Mathematics, 67, Springer-Verlag, New York-Berlin, 1979.Google Scholar
[30] Shalika, J. A., Representations ofthe two by two unimodular group over localfields. Thesis (Ph.D.), The Johns Hopkins University, 1966.Google Scholar
[31] Steinberg, R., Lectures on Chevalley groups. Yale University, New Haven, Conn., 1968.Google Scholar
[32] Weil, A., Sur certains groupes d'operateurs unitaires. Acta Math. 111(1964), 143–211. http://dx.doi.Org/10.1007/BF02391012Google Scholar