×

Construction of tame supercuspidal representations. (English) Zbl 0971.22012

Let \(F\) be a non-archimedean local field. The author constructs a list of supercuspidal representations for any connected reductive linear algebraic group \(G\) that is defined over \(F\) and splits over the maximal tamely ramified extension field of \(F\). The supercuspidal representations on the list are called tame, because the list is expected to be complete if the group is tame, whereas in general the list is incomplete. For \(G=Gl_n\) the tameness of the group means that \((n,p)=1\), where \(p\) is the characteristic of the residual field of \(F\). In general, the author expects that \(G\) is tame (i.e. the list is complete), if \(p\) is large enough.
The construction depends on a set of data \((\vec G,\pi_0, \vec\varphi)\), where the components \(\vec G=(G^0, \dots,G^d)\) and \(\vec\varphi =(\varphi_0, \dots, \varphi_d)\) are generalizations of the tower of generic characters as first defined in the work [J. Algebra 131, 388-424 (1990; Zbl 0715.22019)] of R. Howe and A. Moy for tame \(Gl_n\). The datum \(\pi_0\) is a supercuspidal representation of \(G_0\) of depth 0 and \(G_0\) is an algebraic subgroup of \(G\). All irreducible supercuspidal representations of depth 0 are induced from a compact mod center subgroup [L. Morris, Compos. Math. 118, 135-157 (1999; Zbl 0937.22011); A. Moy and G. Prasad, Comment. Math. Helv. 71, 98-121 (1996; Zbl 0860.22006)]. The author constructs from such an inducing representation and the other data an inducing representation of the associated tame supercuspidal irreducible representation \(\pi=\pi(\vec G,\pi_0,\vec \varphi)\). Finally, the author weakens the assumptions on the data and proves a generalization of a Hecke algebra isomorphism of R. Howe and A. Moy [op. cit.]. This can be considered also in the context of the covering type due to C. J. Bushnell and Ph. C. Kutzko [Proc. Lond. Math. Soc. (3) 77, 582-634 (1998; Zbl 0911.22014)].

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
20G25 Linear algebraic groups over local fields and their integers
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Jeffrey D. Adler, Refined anisotropic \?-types and supercuspidal representations, Pacific J. Math. 185 (1998), no. 1, 1 – 32. · Zbl 0924.22015 · doi:10.2140/pjm.1998.185.1
[2] J.D. Adler and A. Roche: An intertwining result for \(p\)-adic groups, Canad. J. Math. 52, no. 3, 449-467 (2000) CMP 2000:12
[3] Colin J. Bushnell and Philip C. Kutzko, The admissible dual of \?\?(\?) via compact open subgroups, Annals of Mathematics Studies, vol. 129, Princeton University Press, Princeton, NJ, 1993. · Zbl 0787.22016
[4] Colin J. Bushnell and Philip C. Kutzko, Smooth representations of reductive \?-adic groups: structure theory via types, Proc. London Math. Soc. (3) 77 (1998), no. 3, 582 – 634. · Zbl 0911.22014 · doi:10.1112/S0024611598000574
[5] F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5 – 251 (French). · Zbl 0254.14017
[6] F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197 – 376 (French). · Zbl 0597.14041
[7] Lawrence Corwin, A construction of the supercuspidal representations of \?\?_{\?}(\?),\?\?-adic, Trans. Amer. Math. Soc. 337 (1993), no. 1, 1 – 58. · Zbl 0789.22032
[8] S. DeBacker, On supercuspidal characters of \(\operatorname{GL}_\ell\), \(\ell\) a prime, Ph. D. thesis, The University of Chicago (1997)
[9] Paul Gérardin, Weil representations associated to finite fields, J. Algebra 46 (1977), no. 1, 54 – 101. · Zbl 0359.20008 · doi:10.1016/0021-8693(77)90394-5
[10] Roger E. Howe, Tamely ramified supercuspidal representations of \?\?_{\?}, Pacific J. Math. 73 (1977), no. 2, 437 – 460. · Zbl 0404.22019
[11] Roger Howe and Allen Moy, Hecke algebra isomorphisms for \?\?(\?) over a \?-adic field, J. Algebra 131 (1990), no. 2, 388 – 424. · Zbl 0715.22019 · doi:10.1016/0021-8693(90)90182-N
[12] James E. Humphreys, Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975. Graduate Texts in Mathematics, No. 21. · Zbl 0325.20039
[13] J. Kim, Hecke algebras of classical groups over \(p\)-adic fields and supercuspidal representations, Amer. J. Math. 121, 967-1029 (1999) CMP 2000:01
[14] Helmut Koch and Ernst-Wilhelm Zink, Zur Korrespondenz von Darstellungen der Galoisgruppen und der zentralen Divisionsalgebren über lokalen Körpern (der zahme Fall), Math. Nachr. 98 (1980), 83 – 119 (German). · Zbl 0475.12024 · doi:10.1002/mana.19800980110
[15] George Lusztig, Classification of unipotent representations of simple \?-adic groups, Internat. Math. Res. Notices 11 (1995), 517 – 589. · Zbl 0872.20041 · doi:10.1155/S1073792895000353
[16] Lawrence Morris, Level zero \?-types, Compositio Math. 118 (1999), no. 2, 135 – 157. · Zbl 0937.22011 · doi:10.1023/A:1001019027614
[17] Lawrence Morris, Some tamely ramified supercuspidal representations of symplectic groups, Proc. London Math. Soc. (3) 63 (1991), no. 3, 519 – 551. · Zbl 0746.22013 · doi:10.1112/plms/s3-63.3.519
[18] Lawrence Morris, Tamely ramified supercuspidal representations of classical groups. I. Filtrations, Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 6, 705 – 738. Lawrence Morris, Tamely ramified supercuspidal representations of classical groups. II. Representation theory, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 3, 233 – 274. · Zbl 0756.20006
[19] Lawrence Morris, Tamely ramified intertwining algebras, Invent. Math. 114 (1993), no. 1, 1 – 54. · Zbl 0854.22022 · doi:10.1007/BF01232662
[20] Allen Moy, Local constants and the tame Langlands correspondence, Amer. J. Math. 108 (1986), no. 4, 863 – 930. · Zbl 0597.12019 · doi:10.2307/2374518
[21] Allen Moy and Gopal Prasad, Unrefined minimal \?-types for \?-adic groups, Invent. Math. 116 (1994), no. 1-3, 393 – 408. · Zbl 0804.22008 · doi:10.1007/BF01231566
[22] Allen Moy and Gopal Prasad, Jacquet functors and unrefined minimal \?-types, Comment. Math. Helv. 71 (1996), no. 1, 98 – 121. · Zbl 0860.22006 · doi:10.1007/BF02566411
[23] G. Prasad, Galois-fixed points in the Bruhat-Tits building of a reductive group, Bulletin Soc. Math. France, to appear. · Zbl 0992.20032
[24] Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. · Zbl 0355.20006
[25] T. A. Springer, Reductive groups, Automorphic forms, representations and \?-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3 – 27.
[26] Robert Steinberg, Torsion in reductive groups, Advances in Math. 15 (1975), 63 – 92. · Zbl 0312.20026 · doi:10.1016/0001-8708(75)90125-5
[27] J. Tits, Reductive groups over local fields, Automorphic forms, representations and \?-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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.