Hostname: page-component-6bf8c574d5-956mj Total loading time: 0 Render date: 2025-02-17T00:55:46.547Z Has data issue: false hasContentIssue false
  • English
  • Français

Explicit asymptotic expansions for tame supercuspidal characters

Part of: Lie groups

Published online by Cambridge University Press:  12 October 2018

Loren Spice
Loren Spice*
Affiliation:
2840 W. Bowie St, Texas Christian University, Fort Worth, TX 76129, USA email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We combine the ideas of a Harish-Chandra–Howe local character expansion, which can be centred at an arbitrary semisimple element, and a Kim–Murnaghan asymptotic expansion, which so far has been considered only around the identity. We show that, for most smooth, irreducible representations (those containing a good, minimal K-type), Kim–Murnaghan-type asymptotic expansions are valid on explicitly defined neighbourhoods of nearly arbitrary semisimple elements. We then give an explicit, inductive recipe for computing the coefficients in an asymptotic expansion for a tame supercuspidal representation. The only additional information needed in the inductive step is a fourth root of unity, which we expect to be useful in proving stability and endoscopic-transfer identities.

Keywords

p-adic groupsupercuspidal representationcharacter computation

MSC classification

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields 22E35: Analysis on $p$-adic Lie groups
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 11 , November 2018 , pp. 2305 - 2378
Copyright
© The Author 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The author was partially supported by Simons Foundation Collaboration Grant 246066.

References

Adler, J. D., Refined anisotropic K-types and supercuspidal representations , Pacific J. Math. 185 (1998), 1–32; MR 1653184 (2000f:22019).Google Scholar
Adler, J. D. and DeBacker, S., Some applications of Bruhat–Tits theory to harmonic analysis on the Lie algebra of a reductive p-adic group, with appendices by Reid Huntsinger and Gopal Prasad , Michigan Math. J. 50 (2002), 263–286; MR 1914065 (2003g:22016).Google Scholar
Adler, J. D. and Korman, J., The local character expansion near a tame, semisimple element , Amer. J. Math. 129 (2007), 381–403.Google Scholar
Adler, J. D. and Spice, L., Good product expansions for tame elements of p-adic groups , Int. Math. Res. Pap. 2008 (2008), doi:10.1093/imrp/rpn003.Google Scholar
Adler, J. D. and Spice, L., Supercuspidal characters of reductive p-adic groups , Amer. J. Math. 131 (2009), 1136–1210.Google Scholar
Bernšteĭn, I. N. and Zelevinskiĭ, A. V., Representations of the group GL(n, F), where F is a local non-Archimedean field , Uspekhi Mat. Nauk 31 (1976), 5–70 (Russian); MR 0425030 (54 #12988).Google Scholar
Bruhat, F. and Tits, J., Groupes réductifs sur un corps local , Publ. Math. Inst. Hautes Études Sci. 41 (1972), 5–251 (French); MR 0327923 (48 #6265).Google Scholar
Bruhat, F. and Tits, J., Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée , Publ. Math. Inst. Hautes Études Sci. 60 (1984), 197–376 (French); MR 756316 (86c:20042).Google Scholar
Bushnell, C. J. and Kutzko, P. C., The admissible dual of GL(N) via compact open subgroups, Annals of Mathematics Studies, vol. 129 (Princeton University Press, Princeton, NJ, 1993); MR 1204652 (94h:22007).Google Scholar
Casselman, W., Characters and Jacquet modules , Math. Ann. 230 (1977), 101–105; MR 0492083 (58 #11237).Google Scholar
DeBacker, S., Homogeneity results for invariant distributions of a reductive p-adic group , Ann. Sci. Éc. Norm. Supér. (4) 35 (2002), 391–422 (English, with English and French summaries); MR 1914003 (2003i:22019).Google Scholar
DeBacker, S. and Reeder, M., Depth-zero supercuspidal L-packets and their stability , Ann. Math. 169 (2009), 795–901.Google Scholar
DeBacker, S. and Spice, L., Stability of character sums for positive-depth, supercuspidal representations , J. Reine Angew. Math. 2016, https://doi.org/10.1515/crelle-2015-0094.Google Scholar
Deligne, P., Le support du caractère d’une représentation supercuspidale , C. R. Acad. Sci. Paris Sér. A-B 283 (1976), Aii, A155–A157 (French, with English summary); MR 0425033 (54 #12991).Google Scholar
Digne, F. and Michel, J., Groupes réductifs non connexes , Ann. Sci. Éc. Norm. Supér. (4) 27 (1994), 345–406 (French, with English and French summaries); MR 1272294.Google Scholar
Harish-Chandra, Harmonic analysis on reductive p-adic groups, notes by G. van Dijk, Lecture Notes in Mathematics, vol. 162(Springer, Berlin, 1970); MR 0414797 (54 #2889).Google Scholar
Harish-Chandra, A submersion principle and its applications , in Geometry and analysis: Papers dedicated to the memory of V. K. Patodi (Indian Academy of Sciences, Bangalore, 1980), 95–102; MR 592255 (82e:22032).Google Scholar
Harish-Chandra, Admissible invariant distributions on reductive p-adic groups, with a preface and notes by Stephen DeBacker and Paul J. Sally, Jr. , University Lecture Series, vol. 16 (American Mathematical Society, Providence, RI, 1999); MR 1702257 (2001b:22015).Google Scholar
Kaletha, T., Simple wild L-packets , J. Inst. Math. Jussieu 12 (2013), 43–75, doi:10.1017/S1474748012000631; MR 3001735.Google Scholar
Kaletha, T., Epipelagic L-packets and rectifying characters , Invent. Math. 202 (2015), 1–89, doi:10.1007/s00222-014-0566-4.Google Scholar
Kaletha, T., Regular supercuspidal representations, Preprint (2016), arXiv:1602.03144 [math.RT].Google Scholar
Kempf, G. R., Instability in invariant theory , Ann. of Math. (2) 108 (1978), 299–316; MR 506989 (80c:20057).Google Scholar
Kim, J.-L. and Murnaghan, F., Character expansions and unrefined minimal K-types , Amer. J. Math. 125 (2003), 1199–1234; MR 2018660 (2004k:22024).Google Scholar
Kim, J.-L. and Murnaghan, F., K-types and 𝛤-asymptotic expansions , J. Reine Angew. Math. 592 (2006), 189–236.Google Scholar
Kottwitz, R. E., Reductive groups, epsilon factors and Weil indices, Preprint (2016), arXiv:1612.05550 [math.RT].Google Scholar
Mœglin, C. and Waldspurger, J.-L., Modèles de Whittaker dégénérés pour des groupes p-adiques , Math. Z. 196 (1987), 427–452 (French); MR 913667 (89f:22024).Google Scholar
Moy, A. and Prasad, G., Unrefined minimal K-types for p-adic groups , Invent. Math. 116 (1994), 393–408; MR 1253198 (95f:22023).Google Scholar
Moy, A. and Prasad, G., Jacquet functors and unrefined minimal K-types , Comment. Math. Helv. 71 (1996), 98–121; MR 1371680 (97c:22021).Google Scholar
Ranga Rao, R., Orbital integrals in reductive groups , Ann. of Math. (2) 96 (1972), 505–510; MR 0320232 (47 #8771).Google Scholar
Reeder, M., Supercuspidal L-packets of positive depth and twisted Coxeter elements , J. Reine Angew. Math. 620 (2008), 1–33; MR 2427973.Google Scholar
Rodier, F., Intégrabilité locale des caractères du groupe GL(n, k) où k est un corps local de caractéristique positive , Duke Math. J. 52 (1985), 771–792 (French); MR 808104 (87e:22042).Google Scholar
Spice, L., Topological Jordan decompositions , J. Algebra 319 (2008), 3141–3163.Google Scholar
Springer, T. A., Linear algebraic groups , Progress in Mathematics, vol. 9 (Birkhäuser, Boston, 1998); MR 1642713 (99h:20075).Google Scholar
Steinberg, R., Endomorphisms of linear algebraic groups , Memoirs of the American Mathematical Society, No. 80 (American Mathematical Society, Providence, RI, 1968), 108pp; MR 0230728 (37 #6288).Google Scholar
Waldspurger, J.-L., Une formule des traces locale pour les algèbres de Lie p-adiques , J. Reine Angew. Math. 465 (1995), 41–99 (French); MR 1344131 (96i:22039).Google Scholar
Waldspurger, J.-L., Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés , Astérisque 269 (French); MR 1817880 (2002h:22014).Google Scholar
Weil, A., Sur certains groupes d’opérateurs unitaires , Acta Math. 111 (1964), 143–211 (French); MR 0165033 (29 #2324).Google Scholar
Yu, J.-K., Construction of tame supercuspidal representations , J. Amer. Math. Soc. 14 (2001), 579–622 (electronic); MR 1824988 (2002f:22033).Google Scholar
Yu, J.-K., The construction of supercuspidals/types revisited, Talk at conference ‘Characters, liftings, and types’, American University (June 2012).Google Scholar
Yu, J.-K., Smooth models associated to concave functions in Bruhat–Tits theory , in Autour des schémas en groupes. Vol. III, Panor. Synthèses, vol. 47 (Société Mathématique de France, Paris, 2015), 227–258 (English, with English and French summaries); MR 3525846.Google Scholar