Categorical Langlands correspondence for 𝑆𝑂_{𝑛,1}(ℝ)

Halupczok, Immanuel

doi:10.1090/S1088-4165-06-00290-1

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

PDF

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

Combinatorics for the representation theory of real reductive groups

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

Kategorielle Langlands-Korrespondenz für die verallgemeinerten Lorentz-Gruppen

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

Irreducible representations of Lorentz groups

15

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