## Unitary Shimura correspondences for split real groups

HTML articles powered by AMS MathViewer

- by J. Adams, D. Barbasch, A. Paul, P. Trapa and D. A. Vogan Jr. PDF
- J. Amer. Math. Soc.
**20**(2007), 701-751

## Abstract:

We find a relationship between certain complementary series representations for nonlinear coverings of split simple groups, and spherical complementary series for (different) linear groups. The main technique is Barbasch’s method of calculating some intertwining operators purely in terms of the Weyl group.## References

- Jeffrey Adams,
*Nonlinear covers of real groups*, Int. Math. Res. Not.**75**(2004), 4031–4047. MR**2112326**, DOI 10.1155/S1073792804141329 - Jeffrey Adams and Dan Barbasch,
*Genuine representations of the metaplectic group*, Compositio Math.**113**(1998), no. 1, 23–66. MR**1638210**, DOI 10.1023/A:1000450504919 - Dan Barbasch and David A. Vogan Jr.,
*The local structure of characters*, J. Functional Analysis**37**(1980), no. 1, 27–55. MR**576644**, DOI 10.1016/0022-1236(80)90026-9 - D. Barbasch, Spherical unitary dual of split classical groups, preprint. Available at http://www.math.cornell.edu/~barbasch/.
- Dan Barbasch,
*Relevant and petite $K$-types for split groups*, Functional analysis VIII, Various Publ. Ser. (Aarhus), vol. 47, Aarhus Univ., Aarhus, 2004, pp. 35–71. MR**2127163** - Nicolas Bourbaki,
*Éléments de mathématique*, Masson, Paris, 1981 (French). Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6]. MR**647314** - 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** - Dan Ciubotaru,
*The unitary $\Bbb I$-spherical dual for split $p$-adic groups of type $F_4$*, Represent. Theory**9**(2005), 94–137. MR**2123126**, DOI 10.1090/S1088-4165-05-00206-2 - Charles W. Curtis and Irving Reiner,
*Methods of representation theory. Vol. I*, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR**632548** - Roger Howe,
*Wave front sets of representations of Lie groups*, Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., vol. 10, Tata Institute of Fundamental Research, Bombay, 1981, pp. 117–140. MR**633659** - Roger Howe,
*On a notion of rank for unitary representations of the classical groups*, Harmonic analysis and group representations, Liguori, Naples, 1982, pp. 223–331. MR**777342** - Roger Howe,
*Transcending classical invariant theory*, J. Amer. Math. Soc.**2**(1989), no. 3, 535–552. MR**985172**, DOI 10.1090/S0894-0347-1989-0985172-6 - 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 - Jing-Song Huang,
*Metaplectic correspondences and unitary representations*, Compositio Math.**80**(1991), no. 3, 309–322. MR**1134258** - Anthony W. Knapp,
*Representation theory of semisimple groups*, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR**855239**, DOI 10.1515/9781400883974 - Jian-Shu Li,
*Singular unitary representations of classical groups*, Invent. Math.**97**(1989), no. 2, 237–255. MR**1001840**, DOI 10.1007/BF01389041 - Jian-Shu Li,
*On the classification of irreducible low rank unitary representations of classical groups*, Compositio Math.**71**(1989), no. 1, 29–48. MR**1008803** - J.-S. Li, Unipotent representations attached to small nilpotent orbits, unpublished lecture notes from a 1997 AMS conference in Seattle, available at http://www.math.umd./~jda/seattle_proceedings/.
- Jian-Shu Li,
*On the singular rank of a representation*, Proc. Amer. Math. Soc.**106**(1989), no. 2, 567–571. MR**961413**, DOI 10.1090/S0002-9939-1989-0961413-8 - Hideya Matsumoto,
*Sur les sous-groupes arithmétiques des groupes semi-simples déployés*, Ann. Sci. École Norm. Sup. (4)**2**(1969), 1–62 (French). MR**240214**, DOI 10.24033/asens.1174 - Tomasz Przebinda,
*The duality correspondence of infinitesimal characters*, Colloq. Math.**70**(1996), no. 1, 93–102. MR**1373285**, DOI 10.4064/cm-70-1-93-102 - Susana Salamanca-Riba,
*On the unitary dual of some classical Lie groups*, Compositio Math.**68**(1988), no. 3, 251–303. MR**971329** - Gordan Savin,
*Local Shimura correspondence*, Math. Ann.**280**(1988), no. 2, 185–190. MR**929534**, DOI 10.1007/BF01456050 - Gordan Savin,
*On unramified representations of covering groups*, J. Reine Angew. Math.**566**(2004), 111–134. MR**2039325**, DOI 10.1515/crll.2004.001 - Wilfried Schmid,
*On the characters of the discrete series. The Hermitian symmetric case*, Invent. Math.**30**(1975), no. 1, 47–144. MR**396854**, DOI 10.1007/BF01389847 - Robert Steinberg,
*Générateurs, relations et revêtements de groupes algébriques*, Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962) Librairie Universitaire, Louvain; Gauthier-Villars, Paris, 1962, pp. 113–127 (French). MR**0153677** - 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 - David A. Vogan Jr.,
*The unitary dual of $\textrm {GL}(n)$ over an Archimedean field*, Invent. Math.**83**(1986), no. 3, 449–505. MR**827363**, DOI 10.1007/BF01394418 - David A. Vogan Jr.,
*The unitary dual of $G_2$*, Invent. Math.**116**(1994), no. 1-3, 677–791. MR**1253210**, DOI 10.1007/BF01231578 - David A. Vogan Jr.,
*Representations of real reductive Lie groups*, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. MR**632407** - Nolan R. Wallach,
*Real reductive groups. II*, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1992. MR**1170566**

## Additional Information

**J. Adams**- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- Email: jda@math.umd.edu
**D. Barbasch**- Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
- MR Author ID: 30950
- Email: barbasch@math.cornell.edu
**A. Paul**- Affiliation: Department of Mathematics, Western Michigan University, Kalamazoo, Michigan 49008
- Email: paula@wmich.edu
**P. Trapa**- Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
- Email: ptrapa@math.utah.edu
**D. A. Vogan Jr.**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02138
- Email: dav@math.mit.edu
- Received by editor(s): September 6, 2005
- Published electronically: April 11, 2006
- Additional Notes: The first author was supported in part by NSF grant 0532393

The second author was supported in part by NSF grants 0070561 and 0300172

The fourth author was supported in part by NSF grant 0300106

The fifth author was supported in part by NSF grants 9721441 and 0532088

This work began during a visit in 2002 to the Institute for Mathematical Sciences, National University of Singapore. The visit was supported by the Institute and the National University of Singapore. We are grateful to our colleagues at NUS for their generous hospitality. - © Copyright 2006 by J. Adams, D. Barbasch, A. Paul, P. Trapa, and D. A. Vogan, Jr.
- Journal: J. Amer. Math. Soc.
**20**(2007), 701-751 - MSC (2000): Primary 22E46
- DOI: https://doi.org/10.1090/S0894-0347-06-00530-3
- MathSciNet review: 2291917