Representations of reductive groups over finite local rings of length two

Abstract

Let Fq be a finite field of characteristic p, and let W 2 (Fq) be the ring of Witt vectors of length two over Fq. We prove that for any integer n such that p divides n, the groups SLn(Fq[t]/t 2 ) and SLn(W 2 (Fq)) have the same number of irreducible representations of dimension d, for each d.

Enumerating conjugacy classes of graphical groups over finite fields

“…The ‘dual’ problem of enumerating characters (see Section 1.6) is related to recent research developments. In particular, the character theory of reductive groups over rings of the form $$\mathfrak {O}/\mathfrak {P}^2$$ has received considerable attention; see, for example, [30, 33].…”

The $\ell$-modular representation of reductive groups over finite local rings of length two

“…Later on, Stasinski proved it for SL n for all p [14]. More generally, for a possibly non-classical reductive group, Stasinski and Vera-Gajardo compared the complex representation of the two groups in question [15].…”

Representations of SL over finite local rings of length two

“…Let O be a complete discrete valuation ring with maximal ideal p and residue field F q with q elements and characteristic p. For an integer r ≥ 1, we write O r = O/p r . It is known that O 2 , and in fact any finite local ring of length two with residue field F q , is isomorphic to either the ring W 2 (F q ) of Witt vectors of length two or F q [t]/t 2 (see [10,Lemma 2.1]). For a finite group G and an integer d ≥ 1, let Irr d (G) denote the set of isomorphism classes of irreducible complex representations of G of dimension d. P. Singla [9] has proved that when p does not divide n, we have…”

“…In [10] a new proof of this was given, as well as a generalisation when SL n is replaced by any reductive group scheme G over Z (with connected fibres) such that p is very good for G × Z F q . The case G = SL n with p | n was also studied in [9] but the argument there remains incomplete (see [10,Section 5]). In the present paper, we complete the argument and prove that for all n such that p | n, we have…”