Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T03:40:41.409Z Has data issue: false hasContentIssue false
  • English
  • Français

Branching Rules for Principal Series Representations of SL(2) over a p-adic Field

Published online by Cambridge University Press:  20 November 2018

Monica Nevins
Monica Nevins*
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5, e-mail: mnevins@uottawa.ca
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.

We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL\left( 2,\,k \right)$, for $k$ a $p$-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of $SL\left( 2,\,k \right)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup.

Keywords

20G2522E3520H25representations of p-adic groupsp-adic integersorbit methodK-types
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 3 , 01 June 2005 , pp. 648 - 672
Copyright
Copyright © Canadian Mathematical Society 2005

References

[C] Casselman, W., The Restriction of a Representation of GL2(k) to GL2(o). Math. Ann. 206(1973), 311–318.Google Scholar
[DM] Digne, F. and Michel, J., Representations of Finite Groups of Lie Type. LondonMathematical Society Student Texts 21, Cambridge University Press, Cambridge, 1991.Google Scholar
[GGPS] Gel’fand, I. M., Graev, M. I., Pyatetskii-Shapiro, I. I., Representation Theory and Automorphic Functions. Translated from the Russian by Hirsch, K.A.. In: Generalized Functions, 6, Academic Press, Boston, 1990.Google Scholar
[H] Howe, R., Kirillov theory for compact p-adic groups. Pacific J. Math. 73(1977), 365–381.Google Scholar
[K] Kuo, W., Principal nilpotent orbits and reducible principal series. Represent. Theory 6(2002), 127–159.Google Scholar
[LP] Lion, G. et Perrin, P., Extension des représentations de groupes unipotents p-adiques. Calculs d’obstructions. In: Noncommutative harmonic analysis and Lie groups (Marseille, 1980), Lecture Notes in Math. 880, Springer, Berlin, 1981, pp. 337–356.Google Scholar
[R] Roche, A., Types and Hecke algebras for principal series representations of split reductive p-adic groups. Ann. Sci. École Norm. Sup. (4) 31(1998), 361–413.Google Scholar
[S] Serre, J.-P., Linear Representations of Finite Groups. Graduate Texts in Mathematics 42, Springer-Verlag, New York, 1977.Google Scholar
[ShI] Shalika, J. A., Representations of the two by two Unimodular Group over Local Fields, Part I. Seminar on representations of Lie groups. Unpublished notes based on Ph.D. thesis work.Google Scholar
[ShII] Shalika, J. A., Representations of the two by two Unimodular Group over Local Fields, Part II. Seminar on representations of Lie groups. Unpublished notes based on Ph.D. thesis work.Google Scholar
[Si] Silberger, A. J., PGL2 over the p-adics: its representations, spherical functions, and Fourier analysis. Lecture Notes in Mathematics 166, Springer-Verlag, Berlin, 1970.Google Scholar
[Si2] Silberger, A. J., Irreducible representations of a maximal compact subgroup of PGL2 over the p-adics. Math. Ann. 229(1977), 1–12.Google Scholar