×

Types for tame \(p\)-adic groups. (English) Zbl 1492.22013

Let \(k\) stand for a non-Archimedean local field of residual characteristic \(p\), and let us denote by \(G\) a connected reductive group over \(k\), which splits over a tamely ramified extension of \(k\). Let us assume that \(p\) does not divide the order of the Weyl group of \(G.\)
In the paper under the review, the author attaches to an irreducible smooth representation of \(G(k)\) a so-called datum, which is a tuple of the form \[(x, (X_i)_{1 \leq i \leq n}, (\rho_0, V_{\rho_0})),\] where \(n\) is an integer, \(x\) belongs to the Bruhat-Tits building of \(G\), \(X_i \in \mathfrak{g}^{\ast}\), for \(1 \leq i \leq n\), satisfying certain conditions, while \((\rho_0, V_{\rho_0})\) is an irreducible representation of a finite group, obtained as the reductive quotient of the special fiber of the connected parahoric group scheme attached to the derived group of a twisted Levi subgroup of \(G\).
Notion of the datum presents a refinement of the unrefined minimal K-type, introduced by A. Moy and G. Prasad [Invent. Math. 116, No. 1–3, 393–408 (1994; Zbl 0804.22008); Comment. Math. Helv. 71, No. 1, 98–121 (1996; Zbl 0860.22006)]. First main result in the paper is the existence of the maximal datum for any irreducible representation of \(G(k)\), which enables one to produce the input required for the construction of types as in the work of J.-L. Kim and J.-K. Yu [Prog. Math. 323, 337–357 (2017; Zbl 1409.22012)].
This enables the author to prove the second main results of the paper, which states that every smooth irreducible representation of \(G(k)\) contains one of the types constructed by Kim-Yu. Finally, in the third main result, the author uses elements \(X_i\) belonging to the datum to provide appropriate characters of the twisted Levi subgroups and representations \((\rho_0, V_{\rho_0})\) to obtain a depth-zero supercuspidal representations, using the study of Weil representations, which leads to the fact that all irreducible supercuspidal representations of \(G(k)\) can be obtained using the Yu’s construction inducing from open, compact mod center subgroups of \(G(k)\) [J.-K. Yu, J. Am. Math. Soc. 14, No. 3, 579–622 (2001; Zbl 0971.22012)].
We note that the last description of supercuspidal has also been obtained by J.-L. Kim [J. Am. Math. Soc. 20, No. 2, 273–320 (2007; Zbl 1111.22015)], for non-Archimedean local fields of the characteristic zero and under the assumption that \(p\) is very large. On the other hand, proof given in the paper under the review works for non-Archimedean local fields of arbitrary characteristic, with the only requirement that \(p\) does not divide the order of the Weyl group.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Adler, Jeffrey D., Refined anisotropic {\(K\)}-types and supercuspidal representations, Pacific J. Math.. Pacific Journal of Mathematics, 185, 1-32 (1998) · Zbl 0924.22015
[2] Adler, Jeffrey D.; DeBacker, Stephen, Some applications of {B}ruhat-{T}its theory to harmonic analysis on the {L}ie algebra of a reductive {\(p\)}-adic group, Michigan Math. J.. Michigan Mathematical Journal, 50, 263-286 (2002) · Zbl 1018.22013
[3] Adler, Jeffrey D.; Roche, Alan, An intertwining result for {\(p\)}-adic groups, Canad. J. Math.. Canadian Journal of Mathematics. Journal Canadien de Math\'{e}matiques, 52, 449-467 (2000) · Zbl 1160.22304
[4] Bernstein, J. N., Le “centre” de {B}ernstein. Representations of Reductive Groups over a Local Field, Travaux en Cours, 1-32 (1984)
[5] Borel, Armand, Linear Algebraic Groups, Graduate Texts in Math., 126, xii+288 pp. (1991) · Zbl 0726.20030
[6] Borel, Armand; Tits, J., \'{E}l\'{e}ments unipotents et sous-groupes paraboliques de groupes r\'{e}ductifs. {I}, Invent. Math.. Inventiones Mathematicae, 12, 95-104 (1971) · Zbl 0238.20055
[7] Bourbaki, Nicolas, Lie Groups and {L}ie Algebras. {C}hapters 4-6, Elements of Math. (Berlin), xii+300 pp. (2002) · Zbl 0983.17001
[8] Broussous, P., Extension du formalisme de {B}ushnell et {K}utzko au cas d’une alg\`ebre \`a division, Proc. London Math. Soc. (3). Proceedings of the London Mathematical Society. Third Series, 77, 292-326 (1998) · Zbl 0912.22007
[9] Bushnell, Colin J.; Kutzko, Philip C., The Admissible Dual of {\({\rm GL}(N)\)} via Compact Open Subgroups, Annals of Math. Stud., 129, xii+313 pp. (1993) · Zbl 0787.22016
[10] Bushnell, Colin J.; Kutzko, Philip C., The admissible dual of {\({\rm SL}(N)\)}. {II}, Proc. London Math. Soc. (3). Proceedings of the London Mathematical Society. Third Series, 68, 317-379 (1994) · Zbl 0801.22011
[11] Bushnell, Colin J.; Kutzko, Philip C., Smooth representations of reductive {\(p\)}-adic groups: structure theory via types, Proc. London Math. Soc. (3). Proceedings of the London Mathematical Society. Third Series, 77, 582-634 (1998) · Zbl 0911.22014
[12] Bushnell, Colin J.; Kutzko, Philip C., Semisimple types in {\({\rm GL}_n\)}, Compositio Math.. Compositio Mathematica, 119, 53-97 (1999) · Zbl 0933.22027
[13] Carayol, Henri, Repr\'{e}sentations supercuspidales de {\({\rm GL}\sb{n} \)}, C. R. Acad. Sci. Paris S\'{e}r. A-B. Comptes Rendus Hebdomadaires des S\'{e}ances de l’Acad\'{e}mie des Sciences. S\'{e}ries A et B, 288, A17-A20 (1979)
[14] Conrad, Brian, Reductive group schemes. Autour des sch\'{e}mas en groupes. {V}ol. {I}, Panor. Synth\`eses, 42/43, 93-444 (2014) · Zbl 1349.14151
[15] Conrad, Brian; Gabber, Ofer; Prasad, Gopal, Pseudo-Reductive Groups, New Math. Monogr., 26, xxiv+665 pp. (2015) · Zbl 1314.20037
[16] Fintzen, Jessica, On the {M}oy–{P}rasad filtration, {T}o appear in the Journal of the European Mathematical Society (JEMS) (2015)
[17] Fintzen, Jessica, {T}ame tori in \(p\)-adic groups and good semisimple elements, Internat. Math. Res. Not., 22 pp. pp. (2019)
[18] Fintzen, Jessica; Romano, Beth, Stable vectors in {M}oy-{P}rasad filtrations, Compos. Math.. Compositio Mathematica, 153, 358-372 (2017) · Zbl 1373.22024
[19] G{\'{e}}rardin, Paul, Weil representations associated to finite fields, J. Algebra. Journal of Algebra, 46, 54-101 (1977) · Zbl 0359.20008
[20] Goldberg, David; Roche, Alan, Types in {\({\rm SL}_n\)}, Proc. London Math. Soc. (3). Proceedings of the London Mathematical Society. Third Series, 85, 119-138 (2002) · Zbl 1011.22008
[21] Hakim, Jeffrey; Murnaghan, Fiona, Distinguished tame supercuspidal representations, Int. Math. Res. Pap. IMRP. International Mathematics Research Papers. IMRP, 166 pp. pp. (2008) · Zbl 1160.22008
[22] Howe, Roger E., Tamely ramified supercuspidal representations of {\({\rm Gl}\sb{n} \)}, Pacific J. Math.. Pacific Journal of Mathematics, 73, 437-460 (1977) · Zbl 0404.22019
[23] Kaletha, Tasho, Regular supercuspidal representations, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 32, 1071-1170 (2019) · Zbl 1473.22012
[24] Kempf, G. R., Instability in invariant theory, Ann. of Math. (2), 108, 299-316 (1978) · Zbl 0406.14031
[25] Kim, Ju-Lee, Supercuspidal representations: an exhaustion theorem, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 20, 273-320 (2007) · Zbl 1111.22015
[26] Kim, Ju-Lee; Murnaghan, Fiona, Character expansions and unrefined minimal {\(K\)}-types, Amer. J. Math.. American Journal of Mathematics, 125, 1199-1234 (2003) · Zbl 1037.22035
[27] Kim, Ju-Lee; Yu, Jiu-Kang, Construction of tame types. Representation Theory, Number Theory, and Invariant Theory, Progr. Math., 323, 337-357 (2017) · Zbl 1409.22012
[28] Miyauchi, Michitaka; Stevens, Shaun, Semisimple types for {\(p\)}-adic classical groups, Math. Ann.. Mathematische Annalen, 358, 257-288 (2014) · Zbl 1294.22015
[29] Morris, Lawrence, Tamely ramified intertwining algebras, Invent. Math.. Inventiones Mathematicae, 114, 1-54 (1993) · Zbl 0854.22022
[30] Morris, Lawrence, Level zero {\( \bf G\)}-types, Compositio Math.. Compositio Mathematica, 118, 135-157 (1999) · Zbl 0937.22011
[31] Moy, A., Local constants and the tame {L}anglands correspondence, Amer. J. Math., 108, 863-930 (1986) · Zbl 0597.12019
[32] Moy, Allen; Prasad, Gopal, Unrefined minimal {\(K\)}-types for {\(p\)}-adic groups, Invent. Math.. Inventiones Mathematicae, 116, 393-408 (1994) · Zbl 0804.22008
[33] Moy, Allen; Prasad, Gopal, Jacquet functors and unrefined minimal {\(K\)}-types, Comment. Math. Helv.. Commentarii Mathematici Helvetici, 71, 98-121 (1996) · Zbl 0860.22006
[34] Reeder, Mark; Yu, Jiu-Kang, Epipelagic representations and invariant theory, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 27, 437-477 (2014) · Zbl 1284.22011
[35] S\'{e}cherre, V.; Stevens, S., Repr\'{e}sentations lisses de {\({\rm GL}_m(D)\)}. {IV}. {R}epr\'{e}sentations supercuspidales, J. Inst. Math. Jussieu. Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l’Institut de Math\'{e}matiques de Jussieu, 7, 527-574 (2008) · Zbl 1140.22014
[36] S\'{e}cherre, Vincent; Stevens, Shaun, Smooth representations of {\(GL_m(D)\)} {VI}: semisimple types, Int. Math. Res. Not. IMRN. International Mathematics Research Notices. IMRN, 2994-3039 (2012) · Zbl 1246.22023
[37] Springer, T. A.; Steinberg, R., Conjugacy classes. Seminar on {A}lgebraic {G}roups and {R}elated {F}inite {G}roups, Lecture Notes in Math., 131, 167-266 (1970)
[38] Steinberg, Robert, Torsion in reductive groups, Advances in Math.. Advances in Mathematics, 15, 63-92 (1975) · Zbl 0312.20026
[39] Stevens, Shaun, The supercuspidal representations of {\(p\)}-adic classical groups, Invent. Math.. Inventiones Mathematicae, 172, 289-352 (2008) · Zbl 1140.22016
[40] Yu, Jiu-Kang, Construction of tame supercuspidal representations, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 14, 579-622 (2001) · Zbl 0971.22012
[41] Zink, E.-W., Representation theory of local division algebras, J. Reine Angew. Math., 428, 1-44 (1992) · Zbl 0745.22018
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.