On minimal representations definitions and properties

Representation Theory

Volumn 9, Issue 3, 2005, Pages 46-93

  Gan, Wee Teck ^a   Savin, Gordan ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF UTAH   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 29444432211     PISSN: 10884165     EISSN: None     Source Type: Journal    
DOI: 10.1090/S1088-4165-05-00191-3     Document Type: Article

References (39)

1. 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. MR0444849 (56 #3196)
2. R. Brylinski and B. Kostant, Minimal representations, geometric quantization and uni- tarity, Proc. Nat. Acad. Sci. USA 91 (1994), 6026-6029. MR1278630 (95d:58059)
3. R. Carter, Finite Groups of Lie Type. Conjugacy classes and complex characters. John Wiley and Sons, New York, 1985. MR0794307 (87d:20060)
4. W. Casselman, Unramified principal series of p-adic groups I, Compositio Math. 40 (1980), 387-406. MR0571057 (83a:22018)
5. 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. MR0347996 (50 #494)
6. W. T. Gan, A Siegel-Weil formula for exceptional groups, Journal Reine Angew Math. 528 (2000), 149-181. MR1801660 (2001k:11089)
7. W. T. Gan, An automorphic theta module for quaternionic exceptional groups, Canadian J. of Math. 52, Vol. 4 (2000), 737-756. MR1767400 (2001g:11074)
8. 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. MR1918673 (2004e:11051)
9. D. Garfinkle, A new construction of the Joseph ideal, PhD Thesis, MIT.
10. D. Ginzburg, S. Rallis and D. Soudry, On the automorphic theta representation for simply- laced groups, Israel Journal of Math. 100 (1997), 61-116. MR1469105 (99c:11058)
11. 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. MR1486137 (2000i:22024)
12. B. H. Gross and N. Wallach, On quaternionic discrete series representations and their continuations, Journal Reine Angew Math. 481 (1996), 73-123. MR1421947 (98f:22022)
13. J.-S. Huang, Lectures on representation theory, World Scientific Publishing (1999). MR1735324 (2000m:22002)
14. R. Howe and C. Moore, Asymptotic properties of unitary representations, J. Functional Analysis 32 (1979), 72-96. MR0533220 (80g:22017)
15. J.-S. Huang, K. Magaard and G. Savin, Unipotent representations of G2 arising from the minimal representation of DE, Journal Reine Angew Math. 500 (1998), 65-81. MR1637485 (99j:22025)
16. J.-S. Huang, P. Pandzic and G. Savin, New dual pair correspondences, Duke Math. J. 82 (1996), 447-471. MR1387237 (97c:22015)
17. 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. MR0404366 (53 #8168)
18. D. Kazhdan, The minimal representation of D4, in Operator algebras, unitary representations, enveloping algebras and invariant theory; in honor of Jacques Dixmier, Progress in Math. 92 (1990), Birkhäuser, 125-158. MR1103589 (92i:22015)
19. 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. MR1159103 (93f:22019)
20. B. Kostant, Lie group representations on polynomial rings, American J. Math. 85 (1963), 327-404. MR0158024 (28 #1252)
21. J. S. Li, The correspondences of infinitesimal characters for reductive dual pairs in simple Lie groups, Duke Math. J. 97 (1999), 347-377. MR1682229 (2000b:22014)
22. G. Lusztig, Some examples of square integrable representations of semisimple p-adic groups, Trans. Amer. Math. Soc. 277 (1983), 623-653. MR0694380 (84j:22023)
23. K. Magaard and G. Savin, Exceptional ©-correspondences I, Compositio Math. 107 (1997), 89-123. MR1457344 (98i:22015)
24. 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. MR1041060 (91f:11040)
25. C. Moeglin and J.-L. Waldspurger, Modèles de Whittaker dégénérés pour des groupes p-adiques, Math. Z. 196 (1987), 427-452. MR0913667 (89f:22024)
26. 2, Duke Math. J. 90, No. 3 (1997), 465-493. MR1480543 (98k:22073)
27. G. Muić and G. Savin, Complementary series for Hermitian quaternionic groups, Canad. Math. Bull. 43 (2000), 90-99. MR1749954 (2001g:22019)
28. F. Rodier, Décomposition de la série principale des groupes réductifs p-adiques, Lecture Notes in Math. 880 (1981), Springer-Verlag. MR0644842 (83i:22029)
29. K. Rumelhart, Minimal representations of exceptional p-adic groups, Representation Theory 1 (1997), 133-181. MR1455128 (98j:22013)
30. G. Savin, A letter to David Vogan, January 16, 2002.
31. J is the automorphism group of the Jordan algebra, J. Invent. Math. 118 (1994), 141-160. MR1288471 (95i:22017)
32. 2, Journal Reine Angew Math. 434 (1993), 115-126. MR1195692 (94a:22038)
33. 2(k), Canadian Math. Bulletin 42, No. 3 (1999), 393-400. MR703700 (2000h:22011)
34. 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. MR1484858 (99c:22028)
35. 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. MR0546588 (80h:20064)
36. D. Vogan, Singular unitary representations, Lecture Notes in Math. 880 (1981), Springer- Verlag, 506-535. MR0644845 (83k:22036)
37. D. Vogan, The orbit method and primitive ideals for semisimple Lie algebras, Canadian Math. Society Conference Proc. Vol. 5 (1986), 281-316. MR0832204 (87k:17015)
38. M. Weissman, The Fourier-Jacobi map and small representations, Represent. Theory 7 (2003), 275-299. MR1993361 (2004d:22014)
39. A. Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143-211. MR0165033 (29 #2324)