The regular representations of GLN over finite local principal ideal rings

Stasinski, A.; Stevens, S.

Abstract

Let o o be the ring of integers in a non-Archimedean local field with finite residue field, p p its maximal ideal, and r ⩾ 2 r⩾2 an integer. An irreducible representation of the finite group G r = GL N ( o / p r ) Gr=GLN(o/pr), for an integer N ⩾ 2 N⩾2, is called regular if its restriction to the principal congruence kernel K r − 1 = 1 + p r − 1 M N ( o / p r ) Kr−1=1+pr−1MN(o/pr) consists of representations whose stabilisers modulo K 1 K1 are centralisers of regular elements in M N ( o / p ) MN(o/p). The regular representations form the largest class of representations of G r Gr which is currently amenable to explicit construction. Their study, motivated by constructions of supercuspidal representations, goes back to Shintani, but the general case remained open for a long time. In this paper we give an explicit construction of all the regular representations of G r Gr.

Citation

Stasinski, A., & Stevens, S. (2017). The regular representations of GLN over finite local principal ideal rings. Bulletin of the London Mathematical Society, 49(6), 1066-1084. https://doi.org/10.1112/blms.12099

• Journal Article Type: Article
• Acceptance Date: Aug 30, 2017
• Online Publication Date: Oct 19, 2017
• Publication Date: Oct 19, 2017
• Deposit Date: Sep 26, 2017
• Publicly Available Date: Sep 27, 2017
• Journal: Bulletin of the London Mathematical Society
• Print ISSN: 0024-6093
• Electronic ISSN: 1469-2120
• Publisher: Wiley
• Peer Reviewed: Peer Reviewed
• Volume: 49
• Issue: 6
• Pages: 1066-1084
• DOI: https://doi.org/10.1112/blms.12099
• Public URL: https://durham-repository.worktribe.com/output/1375893
• Related Public URLs: https://arxiv.org/abs/1611.04796