On minimal representations definitions and properties

Gan, Wee Teck; Savin, Gordan

doi:10.1090/S1088-4165-05-00191-3

On minimal representations definitions and properties
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by Wee Teck Gan and Gordan Savin

Represent. Theory 9 (2005), 46-93

DOI: https://doi.org/10.1090/S1088-4165-05-00191-3

Published electronically: January 13, 2005

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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

Armand Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976), 233–259. MR 444849, DOI 10.1007/BF01390139
Ranee Brylinski and Bertram Kostant, Minimal representations, geometric quantization, and unitarity, Proc. Nat. Acad. Sci. U.S.A. 91 (1994), no. 13, 6026–6029. MR 1278630, DOI 10.1073/pnas.91.13.6026
Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
W. Casselman, The unramified principal series of ${\mathfrak {p}}$-adic groups. I. The spherical function, Compositio Math. 40 (1980), no. 3, 387–406. MR 571057
C. W. Curtis, N. Iwahori, and R. Kilmoyer, Hecke algebras and characters of parabolic type of finite groups with $(B,$ $N)$-pairs, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 81–116. MR 347996, DOI 10.1007/BF02684695
Wee Teck Gan, A Siegel-Weil formula for exceptional groups, J. Reine Angew. Math. 528 (2000), 149–181. MR 1801660, DOI 10.1515/crll.2000.088
Wee Teck Gan, An automorphic theta module for quaternionic exceptional groups, Canad. J. Math. 52 (2000), no. 4, 737–756. MR 1767400, DOI 10.4153/CJM-2000-031-4
Wee Teck Gan, Nadya Gurevich, and Dihua Jiang, Cubic unipotent Arthur parameters and multiplicities of square integrable automorphic forms, Invent. Math. 149 (2002), no. 2, 225–265. MR 1918673, DOI 10.1007/s002220200210

A new construction of the Joseph ideal

David Ginzburg, Stephen Rallis, and David Soudry, On the automorphic theta representation for simply laced groups, Israel J. Math. 100 (1997), 61–116. MR 1469105, DOI 10.1007/BF02773635
William Graham and David A. Vogan Jr., Geometric quantization for nilpotent coadjoint orbits, Geometry and representation theory of real and $p$-adic groups (Córdoba, 1995) Progr. Math., vol. 158, Birkhäuser Boston, Boston, MA, 1998, pp. 69–137. MR 1486137, DOI 10.1007/978-1-4612-4162-1_{6}
Benedict H. Gross and Nolan R. Wallach, On quaternionic discrete series representations, and their continuations, J. Reine Angew. Math. 481 (1996), 73–123. MR 1421947, DOI 10.1515/crll.1996.481.73
Jing-Song Huang, Lectures on representation theory, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. MR 1735324, DOI 10.1142/9789812815743
Roger E. Howe and Calvin C. Moore, Asymptotic properties of unitary representations, J. Functional Analysis 32 (1979), no. 1, 72–96. MR 533220, DOI 10.1016/0022-1236(79)90078-8
Jing-Song Huang, Kay Magaard, and Gordan Savin, Unipotent representations of $G_2$ arising from the minimal representation of $D_4^E$, J. Reine Angew. Math. 500 (1998), 65–81. MR 1637485, DOI 10.1515/crll.1998.071
Jing-Song Huang, Pavle Pandžić, and Gordan Savin, New dual pair correspondences, Duke Math. J. 82 (1996), no. 2, 447–471. MR 1387237, DOI 10.1215/S0012-7094-96-08220-4
A. Joseph, The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 1, 1–29. MR 404366, DOI 10.24033/asens.1302
David Kazhdan, The minimal representation of $D_4$, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 125–158. MR 1103589
D. Kazhdan and G. Savin, The smallest representation of simply laced groups, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 2, Weizmann, Jerusalem, 1990, pp. 209–223. MR 1159103
Bertram Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404. MR 158024, DOI 10.2307/2373130
Jian-Shu Li, The correspondences of infinitesimal characters for reductive dual pairs in simple Lie groups, Duke Math. J. 97 (1999), no. 2, 347–377. MR 1682229, DOI 10.1215/S0012-7094-99-09715-6
George Lusztig, Some examples of square integrable representations of semisimple $p$-adic groups, Trans. Amer. Math. Soc. 277 (1983), no. 2, 623–653. MR 694380, DOI 10.1090/S0002-9947-1983-0694380-4
K. Magaard and G. Savin, Exceptional $\Theta$-correspondences. I, Compositio Math. 107 (1997), no. 1, 89–123. MR 1457344, DOI 10.1023/A:1000139424441
Colette Mœglin, Marie-France Vignéras, and Jean-Loup Waldspurger, Correspondances de Howe sur un corps $p$-adique, Lecture Notes in Mathematics, vol. 1291, Springer-Verlag, Berlin, 1987 (French). MR 1041060, DOI 10.1007/BFb0082712
C. Mœglin and J.-L. Waldspurger, Modèles de Whittaker dégénérés pour des groupes $p$-adiques, Math. Z. 196 (1987), no. 3, 427–452 (French). MR 913667, DOI 10.1007/BF01200363
Goran Muić, The unitary dual of $p$-adic $G_2$, Duke Math. J. 90 (1997), no. 3, 465–493. MR 1480543, DOI 10.1215/S0012-7094-97-09012-8
Goran Muić and Gordan Savin, Complementary series for Hermitian quaternionic groups, Canad. Math. Bull. 43 (2000), no. 1, 90–99. MR 1749954, DOI 10.4153/CMB-2000-014-5
François Rodier, Décomposition de la série principale des groupes réductifs $p$-adiques, Noncommutative harmonic analysis and Lie groups (Marseille, 1980) Lecture Notes in Math., vol. 880, Springer, Berlin-New York, 1981, pp. 408–424 (French). MR 644842, DOI 10.1007/BFb0090418
Karl E. Rumelhart, Minimal representations of exceptional $p$-adic groups, Represent. Theory 1 (1997), 133–181. MR 1455128, DOI 10.1090/S1088-4165-97-00009-5

A letter to David Vogan

Gordan Savin, Dual pair $G_{\scr J}\times \textrm {PGL}_2$ [where] $G_{\scr J}$ is the automorphism group of the Jordan algebra ${\scr J}$, Invent. Math. 118 (1994), no. 1, 141–160. MR 1288471, DOI 10.1007/BF01231530
Gordan Savin, An analogue of the Weil representation for $G_2$, J. Reine Angew. Math. 434 (1993), 115–126. MR 1195692, DOI 10.1515/crll.1993.434.115
Roger D. Nussbaum, Asymptotic analysis of some functional-differential equations, Dynamical systems, II (Gainesville, Fla., 1981) Academic Press, New York-London, 1982, pp. 277–301. MR 703700, DOI 10.4153/CMB-1999-046-9
Pierre Torasso, Méthode des orbites de Kirillov-Duflo et représentations minimales des groupes simples sur un corps local de caractéristique nulle, Duke Math. J. 90 (1997), no. 2, 261–377 (French). MR 1484858, DOI 10.1215/S0012-7094-97-09009-8
J. Tits, Reductive groups over local fields, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR 546588, DOI 10.1090/pspum/033.1/546588
David A. Vogan Jr., Singular unitary representations, Noncommutative harmonic analysis and Lie groups (Marseille, 1980) Lecture Notes in Math., vol. 880, Springer, Berlin-New York, 1981, pp. 506–535. MR 644845, DOI 10.1007/BFb0090421
David A. Vogan Jr., The orbit method and primitive ideals for semisimple Lie algebras, Lie algebras and related topics (Windsor, Ont., 1984) CMS Conf. Proc., vol. 5, Amer. Math. Soc., Providence, RI, 1986, pp. 281–316. MR 832204, DOI 10.1007/BF01394418
Martin H. Weissman, The Fourier-Jacobi map and small representations, Represent. Theory 7 (2003), 275–299. MR 1993361, DOI 10.1090/S1088-4165-03-00197-3
André Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211 (French). MR 165033, DOI 10.1007/BF02391012