Skip to main content
×
×
Home

INTERTWINING SEMISIMPLE CHARACTERS FOR $p$ -ADIC CLASSICAL GROUPS

  • DANIEL SKODLERACK (a1) and SHAUN STEVENS (a2)
Abstract

Let  $G$ be an orthogonal, symplectic or unitary group over a non-archimedean local field of odd residual characteristic. This paper concerns the study of the “wild part” of an irreducible smooth representation of  $G$ , encoded in its “semisimple character”. We prove two fundamental results concerning them, which are crucial steps toward a complete classification of the cuspidal representations of  $G$ . First we introduce a geometric combinatorial condition under which we prove an “intertwining implies conjugacy” theorem for semisimple characters, both in  $G$ and in the ambient general linear group. Second, we prove a Skolem–Noether theorem for the action of  $G$ on its Lie algebra; more precisely, two semisimple elements of the Lie algebra of  $G$ which have the same characteristic polynomial must be conjugate under an element of  $G$ if there are corresponding semisimple strata which are intertwined by an element of  $G$ .

Copyright
References
Hide All
[BL02] Broussous, P. and Lemaire, B., Building of GL(m, D) and centralizers , Transform. Groups 7(1) (2002), 1550.
[BSS12] Broussous, P., Sécherre, V. and Stevens, S., Smooth representations of GL m (D) V: endo-classes , Doc. Math. 17 (2012), 2377.
[BS09] Broussous, P. and Stevens, S., Buildings of classical groups and centralizers of Lie algebra elements , J. Lie Theory 19(1) (2009), 5578.
[BT87] Bruhat, F. and Tits, J., Schémas en groupes et immeubles des groupes classiques sur un corps local. II. Groupes unitaires , Bull. Soc. Math. France 115(2) (1987), 141195.
[Bus87] Bushnell, C. J., Hereditary orders, Gauss sums and supercuspidal representations of GL N , J. Reine Angew. Math. 375/376 (1987), 184210.
[BH96] Bushnell, C. J. and Henniart, G., Local tame lifting for GL(N). I. Simple characters , Inst. Hautes Études Sci. Publ. Math. 83 (1996), 105233.
[BH03] Bushnell, C. J. and Henniart, G., Local tame lifting for GL(n). IV. Simple characters and base change , Proc. Lond. Math. Soc. (3) 87(2) (2003), 337362.
[BH17] Bushnell, C. J. and Henniart, G., Higher ramification and the local Langlands correspondence , Ann. of Math. (2) 185(3) (2017), 919955.
[BK93] Bushnell, C. J. and Kutzko, P. C., The Admissible Dual of GL(N) Via Compact Open Subgroups, Vol. 129, Princeton University Press, Princeton, NJ, 1993.
[BK94] Bushnell, C. J. and Kutzko, P. C., “ Simple types in GL(N): computing conjugacy classes ”, in Representation Theory and Analysis on Homogeneous Spaces (New Brunswick, NJ, 1993), Contemp. Math. 177 , Amer. Math. Soc., Providence, RI, 1994, 107135.
[BK99] Bushnell, C. J. and Kutzko, P. C., Semisimple types in GL n , Compos. Math. 119(1) (1999), 5397.
[Dat09] Dat, J.-F., Finitude pour les représentations lisses de groupes p-adiques , J. Inst. Math. Jussieu 8(2) (2009), 261333.
[KSS16] Kurinczuk, R., Skodlerack, D. and Stevens, S., Endoclasses for -adic classical groups, preprint, 2016, arXiv:1611.02667.
[KS15] Kurinczuk, R. and Stevens, S., Cuspidal -modular representations of -adic classical groups, preprint, 2015, arXiv:1509.02212.
[Rei03] Reiner, I., Maximal Orders, The Clarendon Press, Oxford University Press, Oxford, 2003.
[SS16] Sécherre, V. and Stevens, S., Towards an explicit local Jacquet–Langlands correspondence beyond the cuspidal case, preprint, 2016, arXiv:1611.04317.
[Sko14] Skodlerack, D., Field embeddings which are conjugate under a p-adic classical group , Manuscripta Math. 144(1–2) (2014), 277301.
[Sko17] Skodlerack, D., Semisimple characters for inner forms I: , preprint, 2017, arXiv:1703.04904.
[Ste01a] Stevens, S., Double coset decompositions and intertwining , Manuscripta Math. 106(3) (2001), 349364.
[Ste01b] Stevens, S., Intertwining and supercuspidal types for p-adic classical groups , Proc. Lond. Math. Soc. (3) 83(1) (2001), 120140.
[Ste02] Stevens, S., Semisimple strata for p-adic classical groups , Ann. Sci. Éc. Norm. Supér. (4) 35(3) (2002), 423435.
[Ste05] Stevens, S., Semisimple characters for p-adic classical groups , Duke Math. J. 127(1) (2005), 123173.
[Ste08] Stevens, S., The supercuspidal representations of p-adic classical groups , Invent. Math. 172(2) (2008), 289352.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed