Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



On minimal representations definitions and properties

Authors: Wee Teck Gan and Gordan Savin
Journal: Represent. Theory 9 (2005), 46-93
MSC (2000): Primary 22E50, and, 22E55
Published electronically: January 13, 2005
MathSciNet review: 2123125
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper gives a self-contained exposition of minimal representations. We introduce a notion of weakly minimal representations and prove a global rigidity result for them. We address issues of uniqueness and existence and prove many key properties of minimal representations needed for global applications.

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

  • [Bo] A. Borel, Admissible representations of a semi-simple group over a local field with fixed vectors under an Iwahori subgroup, Invent Math. 35 (1976), 233-259. MR 0444849 (56 #3196)
  • [BK] R. Brylinski and B. Kostant, Minimal representations, geometric quantization and unitarity, Proc. Nat. Acad. Sci. USA 91 (1994), 6026-6029.MR 1278630 (95d:58059)
  • [Cr] R. Carter, Finite Groups of Lie Type. Conjugacy classes and complex characters. John Wiley and Sons, New York, 1985. MR 0794307 (87d:20060)
  • [Cs] W. Casselman, Unramified principal series of $p$-adic groups I, Compositio Math. 40 (1980), 387-406. MR 0571057 (83a:22018)
  • [CIK] C. W. Curtis, N. Iwahori, and R. Kilmoyer, Hecke algebras and characters of parabolic type of finite groups with $(B,N)$-pairs, Publ. Math. IHES 40 (1971), 81-116. MR 0347996 (50 #494)
  • [G] W. T. Gan, A Siegel-Weil formula for exceptional groups, Journal Reine Angew Math. 528 (2000), 149-181. MR 1801660 (2001k:11089)
  • [G2] W. T. Gan, An automorphic theta module for quaternionic exceptional groups, Canadian J. of Math. 52, Vol. 4 (2000), 737-756. MR 1767400 (2001g:11074)
  • [GGJ] W. T. Gan, N. Gurevich and D. H. Jiang, Cubic unipotent Arthur parameters and multiplicities of square-integrable automorphic forms, Invent. Math. 149 (2002), 225-265. MR 1918673 (2004e:11051)
  • [Ga] D. Garfinkle, A new construction of the Joseph ideal, PhD Thesis, MIT.
  • [GRS] D. Ginzburg, S. Rallis and D. Soudry, On the automorphic theta representation for simply-laced groups, Israel Journal of Math. 100 (1997), 61-116. MR 1469105 (99c:11058)
  • [GV] W. Graham and D. Vogan, Geometric quantization for nilpotent coadjoint orbits, in Geometry and representation theory of real and $p$-adic groups, Progress in Math. 158 (1998), Birkhäuser, 69-137. MR 1486137 (2000i:22024)
  • [GW] B. H. Gross and N. Wallach, On quaternionic discrete series representations and their continuations, Journal Reine Angew Math. 481 (1996), 73-123. MR 1421947 (98f:22022)
  • [H] J.-S. Huang, Lectures on representation theory, World Scientific Publishing (1999). MR 1735324 (2000m:22002)
  • [HM] R. Howe and C. Moore, Asymptotic properties of unitary representations, J. Functional Analysis 32 (1979), 72-96. MR 0533220 (80g:22017)
  • [HMS] J.-S. Huang, K. Magaard and G. Savin, Unipotent representations of $G_2$ arising from the minimal representation of $D_4^E$, Journal Reine Angew Math. 500 (1998), 65-81. MR 1637485 (99j:22025)
  • [HPS] J.-S. Huang, P. Pandzic and G. Savin, New dual pair correspondences, Duke Math. J. 82 (1996), 447-471. MR 1387237 (97c:22015)
  • [J] A. Joseph, The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. Sci. Ecole Norm Sup. 9 (1976), No. 1, 1-29. MR 0404366 (53 #8168)
  • [K] D. Kazhdan, The minimal representation of $D_4$, in Operator algebras, unitary representations, enveloping algebras and invariant theory; in honor of Jacques Dixmier, Progress in Math. 92 (1990), Birkhäuser, 125-158. MR 1103589 (92i:22015)
  • [KS] D. Kazhdan and G. Savin, The smallest representation of simply-laced groups, in Israel Math. Conference Proceedings, Piatetski-Shapiro Festschrift, Vol. 2 (1990), 209-233. MR 1159103 (93f:22019)
  • [Ko] B. Kostant, Lie group representations on polynomial rings, American J. Math. 85 (1963), 327-404. MR 0158024 (28 #1252)
  • [Li] J. S. Li, The correspondences of infinitesimal characters for reductive dual pairs in simple Lie groups, Duke Math. J. 97 (1999), 347-377. MR 1682229 (2000b:22014)
  • [Lu] G. Lusztig, Some examples of square integrable representations of semisimple $p$-adic groups, Trans. Amer. Math. Soc. 277 (1983), 623-653. MR 0694380 (84j:22023)
  • [MS] K. Magaard and G. Savin, Exceptional $\Theta$-correspondences I, Compositio Math. 107 (1997), 89-123. MR 1457344 (98i:22015)
  • [MVW] C. Moeglin, M.-F. Vigneras and J.-L. Waldspurger, Correspondances de Howe sur un corps $p$-adique, Lecture Notes in Math. 1291 (1987), Springer-Verlag. MR 1041060 (91f:11040)
  • [MW] C. Moeglin and J.-L. Waldspurger, Modèles de Whittaker dégénérés pour des groupes $p$-adiques, Math. Z. 196 (1987), 427-452. MR 0913667 (89f:22024)
  • [M] G. Muic, The unitary dual of $p$-adic $G_2$, Duke Math. J. 90, No. 3 (1997), 465-493. MR 1480543 (98k:22073)
  • [MuS] G. Muic and G. Savin, Complementary series for Hermitian quaternionic groups Canad. Math. Bull. 43 (2000), 90-99. MR 1749954 (2001g:22019)
  • [Ro] F. Rodier, Décomposition de la série principale des groupes réductifs $p$-adiques, Lecture Notes in Math. 880 (1981), Springer-Verlag. MR 0644842 (83i:22029)
  • [R] K. Rumelhart, Minimal representations of exceptional $p$-adic groups, Representation Theory 1 (1997), 133-181. MR 1455128 (98j:22013)
  • [S] G. Savin, A letter to David Vogan, January 16, 2002.
  • [S1] G. Savin, Dual pair $G_J \times PGL_2$: $G_J$ is the automorphism group of the Jordan algebra $J$, Invent. Math. 118 (1994), 141-160. MR 1288471 (95i:22017);
  • [S2] G. Savin, An analogue of the Weil representation of $G_2$, Journal Reine Angew Math. 434 (1993), 115-126. MR 1195692 (94a:22038)
  • [S3] G. Savin, A class of supercuspidal representations of $G_2(k)$, Canadian Math. Bulletin 42, No. 3 (1999), 393-400. MR 703700 (2000h:22011)
  • [To] P. Torasso, Méthode des orbites de Kirillov-Duflo et représentations minimales des groupes simples sur un corps local de caracteristisque nulle, Duke Math. Journal 90 (1997), 261-377. MR 1484858 (99c:22028)
  • [Ti] J. Tits, Reductive groups over local fields, in Automorphic Forms, representations and $L$-functions (Corvallis), Proceedings of Symposium in Pure Math. 33 (1979), 29-69. MR 0546588 (80h:20064)
  • [V1] D. Vogan, Singular unitary representations, Lecture Notes in Math. 880 (1981), Springer-Verlag, 506-535. MR 0644845 (83k:22036)
  • [V2] D. Vogan, The orbit method and primitive ideals for semisimple Lie algebras, Canadian Math. Society Conference Proc. Vol. 5 (1986), 281-316. MR 0832204 (87k:17015)
  • [W] M. Weissman, The Fourier-Jacobi map and small representations, Represent. Theory 7 (2003), 275-299. MR 1993361 (2004d:22014)
  • [We] A. Weil, Sur certains groupes d'opérateurs unitaires, Acta Math. 111 (1964), 143-211. MR 0165033 (29 #2324)

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 22E50, and, 22E55

Retrieve articles in all journals with MSC (2000): 22E50, and, 22E55

Additional Information

Wee Teck Gan
Affiliation: Department of Mathematics, University of California San Diego, 9500 Gilman Drive, LaJolla, California 92093-0112

Gordan Savin
Affiliation: Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112-0090

Received by editor(s): March 5, 2003
Received by editor(s) in revised form: April 1, 2004
Published electronically: January 13, 2005
Additional Notes: Wee Teck Gan was partially supported by NSF grant DMS-0202989
Gordan Savin was partially supported by NSF grant DMS-0138604
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society