Categorical Langlands correspondence for $\operatorname {SO}_{n,1}(\mathbb {R})$
HTML articles powered by AMS MathViewer
- by Immanuel Halupczok
- Represent. Theory 10 (2006), 223-253
- DOI: https://doi.org/10.1090/S1088-4165-06-00290-1
- Published electronically: April 6, 2006
Abstract:
In the context of the local Langlands philosopy for $\mathbb {R}$, Adams, Barbasch and Vogan describe a bijection between the simple Harish-Chandra modules for a real reductive group $G(\mathbb {R})$ and the space of “complete geometric parameters”—a space of equivariant local systems on a variety on which the Langlands-dual of $G(\mathbb {R})$ acts. By a conjecture of Soergel, this bijection can be enhanced to an equivalence of categories. In this article, that conjecture is proven in the case where $G(\mathbb {R})$ is a generalized Lorentz group $\operatorname {SO}_{n,1}(\mathbb {R})$.References
- Jeffrey Adams, Dan Barbasch, and David A. Vogan Jr., The Langlands classification and irreducible characters for real reductive groups, Progress in Mathematics, vol. 104, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1162533, DOI 10.1007/978-1-4612-0383-4
- Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527. MR 1322847, DOI 10.1090/S0894-0347-96-00192-0
- Joseph Bernstein and Valery Lunts, Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994. MR 1299527, DOI 10.1007/BFb0073549 FdC Fokko du Cloux, Combinatorics for the representation theory of real reductive groups, Notes from the AIM workshop, Palo Alto, July 2005.
- Mark Goresky, Robert Kottwitz, and Robert MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), no. 1, 25–83. MR 1489894, DOI 10.1007/s002220050197
- Mark Goresky and Robert MacPherson, Intersection homology. II, Invent. Math. 72 (1983), no. 1, 77–129. MR 696691, DOI 10.1007/BF01389130 ID Immanuel Halupczok, Kategorielle Langlands-Korrespondenz für die verallgemeinerten Lorentz-Gruppen, Ph.D. thesis, University of Freiburg, Germany, 2005, http://www.freidok.uni-freiburg.de/volltexte/1715/.
- Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. With a chapter in French by Christian Houzel. MR 1074006, DOI 10.1007/978-3-662-02661-8 Kho S. M. Khoroshkin, Irreducible representations of Lorentz groups, Funct. Anal. Appl. 15 (1981), 114–122.
- George Lusztig and David A. Vogan Jr., Singularities of closures of $K$-orbits on flag manifolds, Invent. Math. 71 (1983), no. 2, 365–379. MR 689649, DOI 10.1007/BF01389103
- R. W. Richardson and T. A. Springer, Combinatorics and geometry of $K$-orbits on the flag manifold, Linear algebraic groups and their representations (Los Angeles, CA, 1992) Contemp. Math., vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 109–142. MR 1247501, DOI 10.1090/conm/153/01309
- Wolfgang Soergel, Langlands’ philosophy and Koszul duality, Algebra—representation theory (Constanta, 2000) NATO Sci. Ser. II Math. Phys. Chem., vol. 28, Kluwer Acad. Publ., Dordrecht, 2001, pp. 379–414. MR 1858045
- T. A. Springer, Linear algebraic groups, Progress in Mathematics, vol. 9, Birkhäuser, Boston, Mass., 1981. MR 632835
- T. A. Springer, The classification of involutions of simple algebraic groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 3, 655–670. MR 927604
- David A. Vogan, Irreducible characters of semisimple Lie groups. III. Proof of Kazhdan-Lusztig conjecture in the integral case, Invent. Math. 71 (1983), no. 2, 381–417. MR 689650, DOI 10.1007/BF01389104
Bibliographic Information
- Immanuel Halupczok
- Affiliation: Albert-Ludwigs-Universität Freiburg, Mathematisches Institut, Eckerstraße 1, 79104 Freiburg, Germany
- Email: math@karimmi.de
- Received by editor(s): June 5, 2005
- Received by editor(s) in revised form: February 6, 2006
- Published electronically: April 6, 2006
- Additional Notes: The author was supported in part by the “Landesgraduiertenförderung Baden-Württemberg”. He also wishes to thank Wolfgang Soergel for making this article possible
- © Copyright 2006 Immanuel Halupczok
- Journal: Represent. Theory 10 (2006), 223-253
- MSC (2000): Primary 22E50, 20G05, 32S60; Secondary 11S37
- DOI: https://doi.org/10.1090/S1088-4165-06-00290-1
- MathSciNet review: 2219113