Transactions of the American Mathematical Society

Author(s): A. G. Helminck; G. F. Helminck
Journal: Trans. Amer. Math. Soc. 350 (1998), 4669-4691.
MSC (1991): Primary 20G15, 20G20, 22E15, 22E46, 53C35
Retrieve article in: PDF
This article is available free of charge

Abstract: Let

be a connected reductive algebraic group defined over a field

of characteristic not 2, $\sigma$ an involution of

defined over

-open subgroup of the fixed point group of $\sigma$ ,

(resp.

) the set of

-rational points of

(resp.

) and

the corresponding symmetric

-variety. A representation induced from a parabolic

-subgroup of

generically contributes to the Plancherel decomposition of

if and only if the parabolic

-subgroup is $\sigma$ -split. So for a study of these induced representations a detailed description of the

-conjucagy classes of these $\sigma$ -split parabolic

-subgroups is needed.

In this paper we give a description of these conjugacy classes for general symmetric

-varieties. This description can be refined to give a more detailed description in a number of cases. These results are of importance for studying representations for real and $\mathfrak p$ -adic symmetric

-varieties.

1.: E. van den Ban, The principal series for a reductive symmetric space I. $H$ -fixed distribution vectors, Ann. Sci. Ec. Norm. Sup. 21 (1988), 359-412. MR 90a:22016
2.: van den Ban, E. and Schlichtkrull, H., The most continuous part of the Plancherel decomposition for a reductive symmetric space, Ann. of Math. (2) 145 (1997), 267-364. CMP 97:10
3.: E. Becker, Valuations and real places in the theory of formally real fields, Géométrie Algébrique Réelle et Formes Quadratiques (Berlin-Heidelberg-New York) (J.-L. Colliot-Thélène, M. Coste, L. Mahé, et M.-F. Roy , ed.), Lecture Notes Mathematics, vol. 959, Springer Verlag, 1982, pp. 1-40. MR 84f:12011
4.: M. Berger, Les espaces symétriques non-compacts, Ann. Sci. École Norm. Sup. 4 (1957), 85-177. MR 21:3516
5.: A. Borel and J. Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55-152. MR 34:7527
6.: -, Compléments a l'article ``groupes réductifs'', Inst. Hautes Études Sci. Publ. Math. 41 (1972), 253-276. MR 47:3556
7.: F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5-252. MR 48:6265
8.: J.-L. Brylinski and P. Delorme, Vecteurs distributions $H$ -invariants pour les séries principales géneralisées d'espaces symétriques réductifs et prolongement méromorphe d'integrales d'Eisenstein, Invent. Math. 109 (1992), 619-664. MR 93m:22016
9.: M. Flensted-Jensen, Discrete series for semisimple symmetric spaces, Annals of Math. 111 (1980), 253-311. MR 81h:22015
10.: Harish-Chandra, Harmonic analysis on real reductive groups I. The theory of the constant term, J. Funct. Anal. 19 (1975), 103-204. MR 53:3201
11.: -, Harmonic analysis on real reductive groups II. Wave packets in the Schwartz space, Invent. Math. 36 (1979), 1-55. MR 55:12874
12.: -, Harmonic analysis on real reductive groups III. The Maass-Selberg relations and the Plancherel formula, Annals of Math. 104 (1976), 117-201. MR 55:12875
13.: -, Harish-Chandra collected papers, Springer-Verlag, New York, 1984. MR 85e:01061
14.: S. Helgason, Differential geometry, Lie groups and symmetric spaces, Pure and Applied Mathematics, vol. XII, Academic Press, New York, 1978. MR 80k:53081
15.: A. G. Helminck, On the classification of symmetric $k$ -varieties I, To appear.
16.: -, Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces, Adv. in Math. 71 (1988), 21-91. MR 90a:17011
17.: -, Tori invariant under an involutorial automorphism I, Adv. in Math. 85 (1991), 1-38. MR 92a:20047
18.: -, Tori invariant under an involutorial automorphism II, Advances in Math. 131 (1997), 1-92. CMP 98:02
19.: -, Tori invariant under an involutorial automorphism III, To appear.
20.: -, On groups with a Cartan involution, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, India), National Board for Higher Mathematics, 1992, pp. 151-192. MR 92j:20042
21.: -, Symmetric $k$ -varieties, Algebraic Groups and Their Generalizations: Classical Methods (Providence, RI), vol. 56, Proc. Sympos. Pure Math., no. Part 1, Amer.Math. Soc, 1994, pp. 233-279. MR 92j:20042
22.: A. G. Helminck and G. F. Helminck, -fixed distributionvectors for representations related to $\mathfrak p$ -adic symmetric varieties, To appear.
23.: A. G. Helminck and S. P. Wang, On rationality properties of involutions of reductive groups, Adv. in Math. 99 (1993), 26-96. MR 94d:20051
24.: J. E. Humphreys, Linear algebraic groups, Springer-Verlag, Berlin, 1975. MR 53:633
25.: H. Jacquet, K. Lai, and S. Rallis, A trace formula for symmetric spaces, Duke Math. J. 70 (1993). MR 94d:11033
26.: G. Lusztig, Symmetric spaces over a finite field, The Grothendieck Festschrift Vol. III (Boston, MA), Progr. Math., vol. 88, Birkhäuser, 1990, pp. 57-81. MR 92e:20034
27.: T. Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979), 331-357. MR 81a:53049
28.: T. Oshima and T. Matsuki, A description of discrete series for semisimple symmetric spaces, Adv. Stud. in Pure Math., vol. 4, Academic Press, Orlando, FL, 1984, pp. 331-390. MR 87m:22042
29.: T. Oshima and J. Sekiguchi, Eigenspaces of invariant differential operators in an affine symmetric space, Invent. Math. 57 (1980), 1-81. MR 81k:43014
30.: A. Prestel, Lectures on formally real fields, Lecture Notes Mathematics, vol. 1093, Springer Verlag, Berlin-Heidelberg-New York, 1984. MR 86h:12013
31.: R.W. Richardson, Orbits, invariants and representations associated to involutions of reductive groups, Invent. Math. 66 (1982), 287-312. MR 83i:14042
32.: W. Rossmann, The structure of semisimple symmetric spaces, Canad. J. Math. 31 (1979), 157-180. MR 81i:53042
33.: I. Satake, Classification theory of semisimple algebraic groups, Lecture Notes in Pure and Appl. Math., vol. 3, Dekker, Berlin, 1971. MR 47:5135
34.: T. A. Springer, Linear algebraic groups, Progr. Math., vol. 9, Birkhäuser, Boston-Basel-Stuttgart, 1981. MR 84i:20002
35.: -, Some results on algebraic groups with involutions, Algebraic groups and related topics, Adv. Stud. in Pure Math., vol. 6, Academic Press, Orlando, FL, 1984, pp. 525-543. MR 86m:20050
36.: J. A. Wolf, Finiteness of orbit structure for real flag manifolds, Geom. Dedicata 3 (1974), 377-384. MR 51:943

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 20G15, 20G20, 22E15, 22E46, 53C35

A. G. Helminck
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina, 27695-8205
Email: loek@math.ncsu.edu

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

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS