A class of parabolic $k$-subgroups associated with symmetric $k$-varieties
HTML articles powered by AMS MathViewer
- by A. G. Helminck and G. F. Helminck PDF
- Trans. Amer. Math. Soc. 350 (1998), 4669-4691 Request permission
Abstract:
Let $G$ be a connected reductive algebraic group defined over a field $k$ of characteristic not 2, $\sigma$ an involution of $G$ defined over $k$, $H$ a $k$-open subgroup of the fixed point group of $\sigma$, $G_k$ (resp. $H_k$) the set of $k$-rational points of $G$ (resp. $H$) and $G_k/H_k$ the corresponding symmetric $k$-variety. A representation induced from a parabolic $k$-subgroup of $G$ generically contributes to the Plancherel decomposition of $L^2(G_k/H_k)$ if and only if the parabolic $k$-subgroup is $\sigma$-split. So for a study of these induced representations a detailed description of the $H_k$-conjucagy classes of these $\sigma$-split parabolic $k$-subgroups is needed. In this paper we give a description of these conjugacy classes for general symmetric $k$-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 $k$-varieties.References
- E. P. van den Ban, The principal series for a reductive symmetric space. I. $H$-fixed distribution vectors, Ann. Sci. Ăcole Norm. Sup. (4) 21 (1988), no. 3, 359â412. MR 974410
- 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), 267â364.
- Eberhard Becker, Valuations and real places in the theory of formally real fields, Real algebraic geometry and quadratic forms (Rennes, 1981) Lecture Notes in Math., vol. 959, Springer, Berlin-New York, 1982, pp. 1â40. MR 683127
- Marcel Berger, Les espaces symĂ©triques noncompacts, Ann. Sci. Ăcole Norm. Sup. (3) 74 (1957), 85â177 (French). MR 0104763
- Armand Borel and Jacques Tits, Groupes rĂ©ductifs, Inst. Hautes Ătudes Sci. Publ. Math. 27 (1965), 55â150 (French). MR 207712
- Armand Borel and Jacques Tits, ComplĂ©ments Ă lâarticle: âGroupes rĂ©ductifsâ, Inst. Hautes Ătudes Sci. Publ. Math. 41 (1972), 253â276 (French). MR 315007
- F. Bruhat and J. Tits, Groupes rĂ©ductifs sur un corps local, Inst. Hautes Ătudes Sci. Publ. Math. 41 (1972), 5â251 (French). MR 327923
- Jean-Luc Brylinski and Patrick Delorme, Vecteurs distributions $H$-invariants pour les sĂ©ries principales gĂ©nĂ©ralisĂ©es dâespaces symĂ©triques rĂ©ductifs et prolongement mĂ©romorphe dâintĂ©grales dâEisenstein, Invent. Math. 109 (1992), no. 3, 619â664 (French). MR 1176208, DOI 10.1007/BF01232043
- Mogens Flensted-Jensen, Discrete series for semisimple symmetric spaces, Ann. of Math. (2) 111 (1980), no. 2, 253â311. MR 569073, DOI 10.2307/1971201
- Harish-Chandra, Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Functional Analysis 19 (1975), 104â204. MR 0399356, DOI 10.1016/0022-1236(75)90034-8
- Harish-Chandra, Harmonic analysis on real reductive groups. II. Wavepackets in the Schwartz space, Invent. Math. 36 (1976), 1â55. MR 439993, DOI 10.1007/BF01390004
- Harish-Chandra, Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula, Ann. of Math. (2) 104 (1976), no. 1, 117â201. MR 439994, DOI 10.2307/1971058
- Alfred Rosenblatt, Sur les points singuliers des Ă©quations diffĂ©rentielles, C. R. Acad. Sci. Paris 209 (1939), 10â11 (French). MR 85
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- A. G. Helminck, On the classification of symmetric $k$-varieties I, To appear.
- Aloysius G. Helminck, Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces, Adv. in Math. 71 (1988), no. 1, 21â91. MR 960363, DOI 10.1016/0001-8708(88)90066-7
- A. G. Helminck, Tori invariant under an involutorial automorphism. I, Adv. Math. 85 (1991), no. 1, 1â38. MR 1087795, DOI 10.1016/0001-8708(91)90048-C
- â, Tori invariant under an involutorial automorphism II, Advances in Math. 131 (1997), 1â92.
- â, Tori invariant under an involutorial automorphism III, To appear.
- A. G. Helminck, On groups with a Cartan involution, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989) Manoj Prakashan, Madras, 1991, pp. 151â192. MR 1131311
- A. G. Helminck, On groups with a Cartan involution, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989) Manoj Prakashan, Madras, 1991, pp. 151â192. MR 1131311
- A. G. Helminck and G. F. Helminck, $H_k$-fixed distributionvectors for representations related to $\mathfrak p$-adic symmetric varieties, To appear.
- A. G. Helminck and S. P. Wang, On rationality properties of involutions of reductive groups, Adv. Math. 99 (1993), no. 1, 26â96. MR 1215304, DOI 10.1006/aima.1993.1019
- James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773
- HervĂ© Jacquet, King F. Lai, and Stephen Rallis, A trace formula for symmetric spaces, Duke Math. J. 70 (1993), no. 2, 305â372. MR 1219816, DOI 10.1215/S0012-7094-93-07006-8
- George Lusztig, Symmetric spaces over a finite field, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, BirkhĂ€user Boston, Boston, MA, 1990, pp. 57â81. MR 1106911, DOI 10.1007/978-0-8176-4576-2_{3}
- Toshihiko Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979), no. 2, 331â357. MR 527548, DOI 10.2969/jmsj/03120331
- Toshio Ćshima and Toshihiko Matsuki, A description of discrete series for semisimple symmetric spaces, Group representations and systems of differential equations (Tokyo, 1982) Adv. Stud. Pure Math., vol. 4, North-Holland, Amsterdam, 1984, pp. 331â390. MR 810636, DOI 10.2969/aspm/00410331
- Toshio Ćshima and Jir\B{o} Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1980), no. 1, 1â81. MR 564184, DOI 10.1007/BF01389818
- Alexander Prestel, Lectures on formally real fields, Lecture Notes in Mathematics, vol. 1093, Springer-Verlag, Berlin, 1984. MR 769847, DOI 10.1007/BFb0101548
- R. W. Richardson, Orbits, invariants, and representations associated to involutions of reductive groups, Invent. Math. 66 (1982), no. 2, 287â312. MR 656625, DOI 10.1007/BF01389396
- W. Rossmann, The structure of semisimple symmetric spaces, Canadian J. Math. 31 (1979), no. 1, 157â180. MR 518716, DOI 10.4153/CJM-1979-017-6
- I. Satake, Classification theory of semi-simple algebraic groups, Lecture Notes in Pure and Applied Mathematics, vol. 3, Marcel Dekker, Inc., New York, 1971. With an appendix by M. Sugiura; Notes prepared by Doris Schattschneider. MR 0316588
- T. A. Springer, Linear algebraic groups, Progress in Mathematics, vol. 9, BirkhÀuser, Boston, Mass., 1981. MR 632835
- T. A. Springer, Some results on algebraic groups with involutions, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 525â543. MR 803346, DOI 10.2969/aspm/00610525
- Joseph A. Wolf, Finiteness of orbit structure for real flag manifolds, Geometriae Dedicata 3 (1974), 377â384. MR 364689, DOI 10.1007/BF00181328
Additional Information
- A. G. Helminck
- Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina, 27695-8205
- Email: loek@math.ncsu.edu
- G. F. Helminck
- Affiliation: Department of Mathematics, Universiteit Twente, Enschede, The Netherlands
- Email: helminck@math.utwente.nl
- Received by editor(s): December 15, 1995
- Received by editor(s) in revised form: December 15, 1996
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 4669-4691
- MSC (1991): Primary 20G15, 20G20, 22E15, 22E46, 53C35
- DOI: https://doi.org/10.1090/S0002-9947-98-02029-7
- MathSciNet review: 1443876