Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Constant term of smooth $H_\psi$ -spherical functions on a reductive -adic group

Author(s): Patrick Delorme
Journal: Trans. Amer. Math. Soc. 362 (2010), 933-955.
MSC (2000): Primary 22E50
Posted: September 17, 2009
MathSciNet review: 2551511
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Let $\psi$ be a smooth character of a closed subgroup, , of a reductive -adic group . If is a parabolic subgroup of such that is open in , we define the constant term of every smooth function on which transforms by $\psi$ under the right action of . The example of mixed models is given: it includes symmetric spaces and Whittaker models. In this case a notion of cuspidal function is defined and studied. It leads to finiteness theorems.

References:

1.: Bernstein, J., Second adjointness theorem for representations of reductive -adic groups, unpublished manuscript.
2.: Benoist, Y, Oh, H., Polar decomposition for -adic symmetric spaces, Int. Math. Res. Not., no. 24, 2007. MR 2377008
3.: Blanc, P., Delorme, P., Vecteurs distributions -invariants de représentations induites, pour un espace symétrique réductif -adique , Ann. Inst. Fourier, 58 (2008) 213-261. MR 2401221
4.: Borel, A., Linear algebraic groups. Second edition. Graduate Texts in Mathematics, 126. Springer-Verlag, New York, 1991. MR 1102012 (92d:20001)
5.: Brylinski, J-L., Delorme, P., Vecteurs distributions -invariants pour les séries principales généralisées d'espaces symétriques réductifs et prolongement méromorphe d'intégrales d'Eisenstein. Invent. Math. 109 (1992) 619-664. MR 1176208 (93m:22016)
6.: Bushnell, C., Representations of reductive -adic groups: Localization of Hecke algebras and applications. J. London Math. Soc., 63 (2001), 364-386. MR 1810135 (2001m:22034)
7.: Bushnell, C., Henniart, G., Generalized Whittaker models and the Bernstein center. Amer. J. Math. 125 (2003), 513-547. MR 1981032 (2005a:22011)
8.: Casselman, W., Introduction to the theory of admissible representations of -adic reductive groups, http://www.math.ubc.ca/ $\sim$ cass/research.html.
9.: Casselman, W., Shalika, J., The unramified principal series of -adic groups. II. The Whittaker function. Compositio Math. 41 (1980), no. 2, 207-231. MR 581582 (83i:22027)
10.: Delorme, P., Espace des coefficients de représentations admissibles d'un groupe réductif -adique, 131-176, Progr. Math., 220, Birkhäuser Boston, Boston, MA, 2004. MR 2036570 (2005c:22025)
11.: Delorme, P., Sécherre, V., An analogue of the Cartan decomposition for -adic reductive symmetric spaces, arXiv:math/0612545.
12.: Harish-Chandra, Harmonic analysis on real reductive groups. I. The theory of the constant term. J. Functional Analysis 19 (1975), 104-204. MR 0399356 (53:3201)
13.: Helminck, A.G., Helminck, G.F., A class of parabolic -subgroups associated with symmetric -varieties. Trans. Amer. Math. Soc. 350 (1998) 4669-4691. MR 1443876 (99g:20082)
14.: Helminck, A. G., Wang, S. P., On rationality properties of involutions of reductive groups. Adv. Math. 99 (1993) 26-96. MR 1215304 (94d:20051)
15.: Kato, S., Takano, K., Subrepresentation theorem for -adic symmetric spaces arXiv:0706.0567, to appear in Int. Math. Res. Not.
16.: Lagier, N., Terme constant de fonctions sur un espace symétrique réductif -adique, J. of Funct. Anal., 254 (2008) 1088-1145. MR 2381204
17.: Offen, O., Sayag, E., Uniqueness and disjointness of Klyachko models, J. Funct. Anal. 254 (2008) 2846-2865. MR 2414223
18.: Tits, J., Reductive groups over local fields. Automorphic forms, representations and -functions, pp. 29-69, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979. MR 546588 (80h:20064)
19.: Waldspurger, J.-L., La formule de Plancherel pour les groupes -adiques (d'après Harish-Chandra), J. Inst. Math. Jussieu 2 (2003), 235-333. MR 1989693 (2004d:22009)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|