Small unitary representations of the double cover of $\operatorname {SL}(m)$
HTML articles powered by AMS MathViewer
- by Adam R. Lucas PDF
- Trans. Amer. Math. Soc. 360 (2008), 3153-3192 Request permission
Abstract:
The irreducible unitary representations of the double cover $\widetilde {\mathrm {SL}(m)}$ of the real group $\mathrm {SL}(m)$, with infinitesimal character $\frac {1}{2}\rho$, which are small in the sense that their annihilator in the universal enveloping algebra is maximal, are expressed as Langlands quotients of generalized principal series. In the case where $m$ is even we show that there are four such representations and in the case where $m$ is odd there is just one. The representations’ smallness allows them to be written as a sum of virtual representations, leading to a character formula for their $K$-types. We investigate the place of these small representations in the orbit method and, in the case of $\widetilde {\mathrm {SL}(2l+1)}$, show that the representation is attached to a nilpotent coadjoint orbit.The $K$-type spectrum for the Langlands quotients is explicitly determined and shown to be multiplicity free.References
- A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk 17 (1962), no. 4 (106), 57–110 (Russian). MR 0142001
- Bertram Kostant, Quantization and unitary representations. I. Prequantization, Lectures in modern analysis and applications, III, Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970, pp. 87–208. MR 0294568
- Michel Duflo, Sur la classification des idéaux primitifs dans l’algèbre enveloppante d’une algèbre de Lie semi-simple, Ann. of Math. (2) 105 (1977), no. 1, 107–120. MR 430005, DOI 10.2307/1971027
- Pierre Torasso, Quantification géométrique, opérateurs d’entrelacement et représentations unitaires de $(\widetilde \textrm {SL})_3(\textbf {R})$, Acta Math. 150 (1983), no. 3-4, 153–242 (French). MR 709141, DOI 10.1007/BF02392971
- Birgit Speh and David A. Vogan Jr., Reducibility of generalized principal series representations, Acta Math. 145 (1980), no. 3-4, 227–299. MR 590291, DOI 10.1007/BF02414191
- Jing-Song Huang, The unitary dual of the universal covering group of $\textrm {GL}(n,\textbf {R})$, Duke Math. J. 61 (1990), no. 3, 705–745. MR 1084456, DOI 10.1215/S0012-7094-90-06126-5
- Jeffrey Adams and Jing-Song Huang, Kazhdan-Patterson lifting for $\textrm {GL}(n,\mathbf R)$, Duke Math. J. 89 (1997), no. 3, 423–444. MR 1470338, DOI 10.1215/S0012-7094-97-08919-5
- David A. Vogan Jr., Irreducible characters of semisimple Lie groups. I, Duke Math. J. 46 (1979), no. 1, 61–108. MR 523602
- David A. Vogan Jr., Gel′fand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), no. 1, 75–98. MR 506503, DOI 10.1007/BF01390063
- David A. Vogan Jr., Associated varieties and unipotent representations, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 315–388. MR 1168491
- David A. Vogan Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. MR 632407
- David A. Vogan Jr., The algebraic structure of the representation of semisimple Lie groups. I, Ann. of Math. (2) 109 (1979), no. 1, 1–60. MR 519352, DOI 10.2307/1971266
- Anthony W. Knapp and David A. Vogan Jr., Cohomological induction and unitary representations, Princeton Mathematical Series, vol. 45, Princeton University Press, Princeton, NJ, 1995. MR 1330919, DOI 10.1515/9781400883936
- A. Joseph, A characteristic variety for the primitive spectrum of a semisimple Lie algebra, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1976), Lecture Notes in Math., Vol. 587, Springer, Berlin, 1977, pp. 102–118. MR 0450350
- Wilfried Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969/70), 61–80 (German). MR 259164, DOI 10.1007/BF01389889
- D. P. Zhelobenko, Kompaktnye gruppy Li i ikh predstavleniya, Izdat. “Nauka”, Moscow, 1970 (Russian). MR 0473097
- David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060
- W. Rossmann, Picard-Lefschetz theory for the coadjoint quotient of a semisimple Lie algebra, Invent. Math. 121 (1995), no. 3, 531–578. MR 1353308, DOI 10.1007/BF01884311
- W. Rossmann, Picard-Lefschetz theory and characters of a semisimple Lie group, Invent. Math. 121 (1995), no. 3, 579–611. MR 1353309, DOI 10.1007/BF01884312
- Sigurdur Helgason, Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, vol. 39, American Mathematical Society, Providence, RI, 1994. MR 1280714, DOI 10.1090/surv/039
- Henryk Hecht and Wilfried Schmid, A proof of Blattner’s conjecture, Invent. Math. 31 (1975), no. 2, 129–154. MR 396855, DOI 10.1007/BF01404112
- Michael Artin, Algebra, Prentice Hall, Inc., Englewood Cliffs, NJ, 1991. MR 1129886
Additional Information
- Adam R. Lucas
- Affiliation: Department of Mathematics, Saint Mary’s College of California, P.O. Box 3517, Moraga, California 94575-3517
- Email: arl3@stmarys-ca.edu
- Received by editor(s): April 11, 2005
- Received by editor(s) in revised form: May 27, 2006
- Published electronically: January 29, 2008
- Additional Notes: The author is thankful to his advisor, Professor David Vogan, for his guidance and endless patience, as well as Peter Trapa and Thom Pietraho for helpful discussions
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 3153-3192
- MSC (2000): Primary 22E46; Secondary 22E15
- DOI: https://doi.org/10.1090/S0002-9947-08-04401-2
- MathSciNet review: 2379792