Typical representations via fixed point sets in Bruhat–Tits buildings

Latham, Peter; Nevins, Monica

doi:10.1090/ert/590

Typical representations via fixed point sets in Bruhat–Tits buildings
HTML articles powered by AMS MathViewer

by Peter Latham and Monica Nevins

Represent. Theory 25 (2021), 1021-1048

DOI: https://doi.org/10.1090/ert/590

Published electronically: December 16, 2021

PDF | Request permission

Abstract:

For a tame supercuspidal representation $\pi$ of a connected reductive $p$-adic group $G$, we establish two distinct and complementary sufficient conditions, formulated in terms of the geometry of the Bruhat–Tits building of $G$, for the irreducible components of its restriction to a maximal compact subgroup to occur in a representation of $G$ which is not inertially equivalent to $\pi$. The consequence is a set of broadly applicable tools for addressing the branching rules of $\pi$ and the unicity of $[G,\pi ]_G$-types.

References

Jeffrey D. Adler, Refined anisotropic $K$-types and supercuspidal representations, Pacific J. Math. 185 (1998), no. 1, 1–32. MR 1653184, DOI 10.2140/pjm.1998.185.1
J. N. Bernstein, Le “centre” de Bernstein, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 1–32 (French). Edited by P. Deligne. MR 771671
Colin J. Bushnell and Philip C. Kutzko, The admissible dual of $\textrm {GL}(N)$ via compact open subgroups, Annals of Mathematics Studies, vol. 129, Princeton University Press, Princeton, NJ, 1993. MR 1204652, DOI 10.1515/9781400882496
Colin J. Bushnell and Philip C. Kutzko, Smooth representations of reductive $p$-adic groups: structure theory via types, Proc. London Math. Soc. (3) 77 (1998), no. 3, 582–634. MR 1643417, DOI 10.1112/S0024611598000574
Corinne Blondel, Quelques propriétés des paires couvrantes, Math. Ann. 331 (2005), no. 2, 243–257 (French). MR 2115455, DOI 10.1007/s00208-004-0579-1
Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822, DOI 10.1007/978-3-642-51438-8
Mikhail Borovoi, Abelian Galois cohomology of reductive groups, Mem. Amer. Math. Soc. 132 (1998), no. 626, viii+50. MR 1401491, DOI 10.1090/memo/0626
F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251 (French). MR 327923, DOI 10.1007/BF02715544
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). MR 756316
William Casselman, The restriction of a representation of $\textrm {GL}_{2}(k)$ to $\textrm {GL}_{2}({\mathfrak {o}})$, Math. Ann. 206 (1973), 311–318. MR 338274, DOI 10.1007/BF01355984
Jessica Fintzen, On the Moy-Prasad filtration, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 12, 4009–4063. MR 4321207, DOI 10.4171/jems/1098
Jessica Fintzen, Types for tame $p$-adic groups, Ann. of Math. (2) 193 (2021), no. 1, 303–346. MR 4199732, DOI 10.4007/annals.2021.193.1.4
Jeffrey Hakim, Constructing tame supercuspidal representations, Represent. Theory 22 (2018), 45–86. MR 3817964, DOI 10.1090/ert/514
Jeffrey Hakim and Fiona Murnaghan, Distinguished tame supercuspidal representations, Int. Math. Res. Pap. IMRP 2 (2008), Art. ID rpn005, 166. MR 2431732
Tasho Kaletha, Regular supercuspidal representations, J. Amer. Math. Soc. 32 (2019), no. 4, 1071–1170. MR 4013740, DOI 10.1090/jams/925
Ju-Lee Kim, Supercuspidal representations: an exhaustion theorem, J. Amer. Math. Soc. 20 (2007), no. 2, 273–320. MR 2276772, DOI 10.1090/S0894-0347-06-00544-3
Robert E. Kottwitz, Isocrystals with additional structure. II, Compositio Math. 109 (1997), no. 3, 255–339. MR 1485921, DOI 10.1023/A:1000102604688
P. C. Kutzko, Mackey’s theorem for nonunitary representations, Proc. Amer. Math. Soc. 64 (1977), no. 1, 173–175. MR 442145, DOI 10.1090/S0002-9939-1977-0442145-3
Ju-Lee Kim and Jiu-Kang Yu, Construction of tame types, Representation theory, number theory, and invariant theory, Progr. Math., vol. 323, Birkhäuser/Springer, Cham, 2017, pp. 337–357. MR 3753917, DOI 10.1007/978-3-319-59728-7_{1}2
Peter Latham, Unicity of types for supercuspidal representations of $p$-adic $\mathbf {SL}_2$, J. Number Theory 162 (2016), 376–390. MR 3448273, DOI 10.1016/j.jnt.2015.10.008
Peter Latham, The unicity of types for depth-zero supercuspidal representations, Represent. Theory 21 (2017), 590–610. MR 3735454, DOI 10.1090/ert/511
Peter Latham, On the unicity of types in special linear groups, Manuscripta Math. 157 (2018), no. 3-4, 445–465. MR 3858412, DOI 10.1007/s00229-018-1006-3
Peter Latham and Monica Nevins, On the unicity of types for toral supercuspidal representations, Representations of reductive $p$-adic groups, Progr. Math., vol. 328, Birkhäuser/Springer, Singapore, 2019, pp. 175–190. MR 3889763, DOI 10.1007/978-981-13-6628-4_{6}
Lawrence Morris, Level zero $\bf G$-types, Compositio Math. 118 (1999), no. 2, 135–157. MR 1713308, DOI 10.1023/A:1001019027614
Allen Moy and Gopal Prasad, Jacquet functors and unrefined minimal $K$-types, Comment. Math. Helv. 71 (1996), no. 1, 98–121. MR 1371680, DOI 10.1007/BF02566411
Ralf Meyer and Maarten Solleveld, Characters and growth of admissible representations of reductive $p$-adic groups, J. Inst. Math. Jussieu 11 (2012), no. 2, 289–331. MR 2905306, DOI 10.1017/S1474748011000120
Santosh Nadimpalli, Typical representations for level zero Bernstein components of $\textrm {GL}_n(F)$, J. Algebra 469 (2017), 1–29. MR 3563005, DOI 10.1016/j.jalgebra.2016.08.033
Santosh Nadimpalli, On classification of typical representations for $\textrm {GL}_3(F)$, Forum Math. 31 (2019), no. 4, 917–941. MR 3975668, DOI 10.1515/forum-2019-0031
Monica Nevins, Branching rules for supercuspidal representations of $\textrm {SL}_2(k)$, for $k$ a $p$-adic field, J. Algebra 377 (2013), 204–231. MR 3008903, DOI 10.1016/j.jalgebra.2012.12.003
Monica Nevins, On branching rules of depth-zero representations, J. Algebra 408 (2014), 1–27. MR 3197168, DOI 10.1016/j.jalgebra.2014.03.016
Vytautas Paskunas, Unicity of types for supercuspidal representations of $\textrm {GL}_N$, Proc. London Math. Soc. (3) 91 (2005), no. 3, 623–654. MR 2180458, DOI 10.1112/S0024611505015340
G. Pappas and M. Rapoport, Twisted loop groups and their affine flag varieties, Adv. Math. 219 (2008), no. 1, 118–198. With an appendix by T. Haines and Rapoport. MR 2435422, DOI 10.1016/j.aim.2008.04.006
V. Sécherre and S. Stevens, Représentations lisses de $\textrm {GL}_m(D)$. IV. Représentations supercuspidales, J. Inst. Math. Jussieu 7 (2008), no. 3, 527–574 (French, with English and French summaries). MR 2427423, DOI 10.1017/S1474748008000078
Shaun Stevens, Semisimple characters for $p$-adic classical groups, Duke Math. J. 127 (2005), no. 1, 123–173. MR 2126498, DOI 10.1215/S0012-7094-04-12714-9
Shaun Stevens, The supercuspidal representations of $p$-adic classical groups, Invent. Math. 172 (2008), no. 2, 289–352. MR 2390287, DOI 10.1007/s00222-007-0099-1
Jiu-Kang Yu, Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14 (2001), no. 3, 579–622. MR 1824988, DOI 10.1090/S0894-0347-01-00363-0
Jiu-Kang Yu, Smooth models associated to concave functions in Bruhat-Tits theory, Autour des schémas en groupes. Vol. III, Panor. Synthèses, vol. 47, Soc. Math. France, Paris, 2015, pp. 227–258 (English, with English and French summaries). MR 3525846