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Durham University

Department of Mathematical Sciences

Academic Staff

Publication details for Alexander Stasinski

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.

Author(s) from Durham

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.