Abstract
Let G be a reductive group over a p-adic field F. Then, as with reductive groups over any field, it is natural to cast the representation theory of G in terms of parabolic induction. This leads to the notion of supercuspidal representation and, in the case of GLn, to the classification of irreducible (admissible) representations given in the work of Bernstein-Zelevinski [BZ], [Z]. On the other hand, the fact that G is a totally disconnected, locally compact group accounts for the existence of open, compact modulo center subgroups of G which in turn has a strong influence on its representation theory. In particular, one is led to consider the possibility that supercuspidal representations may be constructed by induction from such subgroups (see [Ku2] for historical background) and, more generally, to inquire into the possibility of classifying admissible representations of G by considering the subrepresentations they may have when restricted to such subgroups (the possibility of classifying the admissible dual in this fashion was first raised in [H].) In what follows, we report on recent progress in this direction in the case G = GLn(F)\ we begin with some general background.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The first author was supported in part by SERC grant GR/E 47650. The second author was supported in part by NSF Grant DMS-8704194 and by SERC grant GR/F 73366. Borth author wish to thank the Institute for Advanced Study for their hospitality during Academic Year 1988-1989. This visit was supportes in part by NSF Grant DMS 8610730
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive p-adic groups, Ann. Scient. Ec. Norm. Sup. (4) 10 (1977), 441–472.
A. Borel, Admissible representations of a semisimple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976), 233–259.
C. Bushnell, Hereditary orders, Gauss sums, and supercuspidal representations of GL N , J. Reine Angew. Math. 375/376 (1987), 184210.
C. Bushnell and A. Frohlich, Non-abelian congruence Gauss sums and p-adic simple algebras, Proc. London Math. Soc. (3), 50 (1985), 207–264.
H. Carayol, Représentations cuspidales du group linéaire, Ann. Sci. Ecole Norm. Sup. (4) 17 (1984), 191–225.
W. Casselman, Introduction to the theory of admissible representations of p-adic reductive groups, preprint.
R. E. Howe, Some qualitative results on the representation theory ofGL n over a p-adic field, Pac. J. Math. 73 (1977), 497–538.
P. Kutzko, Towards a classification of the supercuspidal representations of GL n , J. London Math. Soc. (2) 37 (1988), 265–274.
P. Kutzko, On the supercuspidal representations of GL N and other reductive groups, Proc. Int. Cong. Math, Berkeley 1986 (AMS 1987), 853–861.
P. Kutzko, On the restriction of supercuspidal representations to compact, open subgroups, Duke Math. J. 52 (1985), 753–764.
P. Kutzko and D. Manderscheid, On intertwining operators for GL N (F), F a non-archimedean local field, Duke Math J. 57 (1988), 275-293.
P. Kutzko and D. Manderscheid, On the supercuspidal representations of GL N , N the product of two primes, Ann. Sci. Ec. Norm. Sup. (4) 23 (1990), 39–88.
J.-L. Waldspurger, Algebres de Heche et induites de representations cuspidales, pour GL(N), J. Reine Angew. Math. 370 (1986), 127–191.
A. V. Zelevinsky, Induced representations of reductive p-adic groups II; On irreducible representations of GL(n), Ann. Scient. Norm. Sup. (4) 13 (1980), 165–210.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer Science+Business Media New York
About this chapter
Cite this chapter
Bushnell, C.J., Kutzko, P.C. (1991). The Admissible Dual of GL N Via Restriction to Compact Opent Subgroups. In: Barker, W.H., Sally, P.J. (eds) Harmonic Analysis on Reductive Groups. Progress in Mathematics, vol 101. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0455-8_5
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0455-8_5
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6768-3
Online ISBN: 978-1-4612-0455-8
eBook Packages: Springer Book Archive