Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the spectrum of $ (Spin(10,2),SL(2,\mathbb{R}))$ in $ E_{7,3}$


Author: Zhe Du
Journal: Proc. Amer. Math. Soc. 139 (2011), 757-768
MSC (2000): Primary 22E46
DOI: https://doi.org/10.1090/S0002-9939-2010-10664-0
Published electronically: September 27, 2010
MathSciNet review: 2736354
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give the complete description of the spectrum of $ (Spin(10,2),SL(2,\mathbb{R}))$ in the minimal representation of $ E_{7,3}$, and we find that it only consists of a discrete spectrum. This is interesting as both groups are noncompact. Also, most of the representations occurring in the spectrum are unitary representations with nonzero cohomology. The representations we obtained have different Hodge types from the representations we obtain from classical theta liftings.


References [Enhancements On Off] (What's this?)

  • 1. Bourbaki, Nicolas, Lie groups and Lie algebras, Chapters 4-6, translated from the 1968 French original by Andrew Pressley. Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. MR 1890629 (2003a:17001)
  • 2. Collingwood, David H.; McGovern, William M., Nilpotent orbits in semisimple Lie algebras. Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York, 1993. MR 1251060 (94j:17001)
  • 3. Enright, Thomas; Howe, Roger; Wallach, Nolan, A classification of unitary highest weight modules. Representation theory of reductive groups (Park City, Utah, 1982), 97-143, Progr. Math., 40, Birkhäuser Boston, Boston, MA, 1983. MR 733809 (86c:22028)
  • 4. Howe, Roger, $ \theta $-series and invariant theory. Automorphic forms, representations and $ L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, pp. 275-285, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, RI, 1979. MR 546602 (81f:22034)
  • 5. Howe, Roger, Reciprocity laws in the theory of dual pairs. Representation theory of reductive groups (Park City, Utah, 1982), 159-175, Progr. Math., 40, Birkhäuser Boston, Boston, MA, 1983. MR 733812 (85k:22033)
  • 6. Howe, Roger, Transcending classical invariant theory. J. Amer. Math. Soc. 2 (1989), No. 3, 535-552. MR 985172 (90k:22016)
  • 7. Howe, Roger, Remarks on classical invariant theory. Trans. Amer. Math. Soc. 313 (1989), No. 2, 539-570. MR 986027 (90h:22015a)
  • 8. Huang, Jing-Song; Pandzic, Palve; Sacin, Gordan, New dual pair correspondence. Duke Math. J. 82 (1996), No. 2, 447-471. MR 1387237 (97c:22015)
  • 9. Hecht, Henryk; Schmid, Wilfried, A proof of Blattner's conjecture. Invent. Math. 31 (1975), No. 2, 129-154. MR 0396855 (53:715)
  • 10. Harris, Michael; Li, Jian-Shu, A Lefschetz property for subvarieties of Shimura varieties. J. Algebraic Geom. 7 (1998), No. 1, 77-122. MR 1620690 (99e:14027)
  • 11. Li, Jian-Shu, The correspondence of infinitesimal characters for reductive dual pairs in simple Lie groups, Duke Mathematics Journal 97 (1999), No. 2, 347-377. MR 1682229 (2000b:22014)
  • 12. Li, Jian-Shu, A description of the discrete spectrum of ( $ SL(2),E_{7(-25)}$), Asian J. Math. 3 (1999), No. 2, 359-372. MR 1796508 (2001m:22029)
  • 13. Li, Jian-Shu, Minimal representations and reductive dual pairs. Representation theory of Lie groups (Park City, UT, 1998), 293-340, IAS/Park City Math. Ser., 8, Amer. Math. Soc., Providence, RI, 2000. MR 1737731 (2001a:22013)
  • 14. Li, Jian-Shu, Two reductive dual pairs in groups of type $ E$. Manuscripta Math. 91 (1996), No. 2, 163-177. MR 1411651 (97j:22037)
  • 15. Li, Jian-Shu, Singular unitary representations of classical groups. Invent. Math. 97 (1989), No. 2, 237-255. MR 1001840 (90h:22021)
  • 16. Li, Jian-Shu, Theta series and construction of automorphic forms. Representation theory of groups and algebras, 237-248, Contemp. Math., 145, Amer. Math. Soc., Providence, RI, 1993. MR 1216192
  • 17. Li, Jian-Shu, Theta lifting for unitary representations with nonzero cohomology. Duke Math. J. 61 (1990), No. 3, 913-937. MR 1084465 (92f:22024)
  • 18. Kudla, Stephen S., Seesaw dual reductive pairs. Automorphic forms of several variables (Katata, 1983), 244-268, Progr. Math., 46, Birkhäuser Boston, Boston, MA, 1984. MR 763017 (86b:22032)
  • 19. Knapp, Anthony W., Lie groups beyond an introduction. Second edition. Progress in Mathematics, 140. Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1920389 (2003c:22001)
  • 20. Knapp, Anthony W. Representation theory of semisimple groups. An overview based on examples. Reprint of the 1986 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 2001. MR 1880691 (2002k:22011)
  • 21. Knapp, Anthony W.; Vogan, David A., Jr., Cohomological induction and unitary representations. Princeton Mathematical Series, 45. Princeton University Press, Princeton, NJ, 1995. MR 1330919 (96c:22023)
  • 22. Kobayashi, Toshiyuki, Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs. Representation theory and automorphic forms, 45-109, Progr. Math., 255, Birkhäuser Boston, Boston, MA, 2008. MR 2369496 (2008m:22024)
  • 23. Rallis, S.; Schiffmann, G., The orbit and theta correspondences for some dual pairs. J. Math. Kyoto Univ. 35 (1995), 429-493. MR 1359007 (96k:22027)
  • 24. Vogan, David A., Jr., Gelfand-Kirillov dimension for Harish-Chandra modules. Invent. Math. 48 (1978), No. 1, 75-98. MR 0506503 (58:22205)
  • 25. Vogan, David A., Jr., The algebraic structure of the representation of semisimple Lie groups. I. Ann. of Math. (2) 109 (1979), No. 1, 1-60. MR 519352 (81j:22020)
  • 26. Vogan, David A., Jr., Singular unitary representations. Noncommutative harmonic analysis and Lie groups (Marseille, 1980), pp. 506-535, Lecture Notes in Math., 880, Springer, Berlin-New York, 1981. MR 644845 (83k:22036)
  • 27. Vogan, David A., Jr., Representations of real reductive Lie groups. Progress in Mathematics, 15. Birkhäuser, Boston, Mass., 1981. MR 632407 (83c:22022)
  • 28. Vogan, David A., Jr.; Zuckerman, Gregg J., Unitary representations with nonzero cohomology. Compositio Math. 53 (1984), No. 1, 51-90. MR 762307 (86k:22040)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 22E46

Retrieve articles in all journals with MSC (2000): 22E46


Additional Information

Zhe Du
Affiliation: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China
Email: duzhe@amss.ac.cn

DOI: https://doi.org/10.1090/S0002-9939-2010-10664-0
Received by editor(s): March 28, 2010
Published electronically: September 27, 2010
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society