[Paper Review] Regular characters of groups of type A_n over discrete valuation rings

Roi Krakovski, Uri Onn+1 more|arXiv (Cornell University)|Apr 4, 2016

Finite Group Theory Research11 references3 citations

TL;DR

This paper constructs regular irreducible representations of general and special linear groups and unitary groups over complete discrete valuation rings with finite residue fields of odd characteristic. It provides explicit uniform formulae for the degrees of these representations and computes their representation zeta functions, establishing Ennola duality and extending results to the special linear and unitary cases via Lie algebra characters and group-theoretic descent.

ABSTRACT

Let O be a complete discrete valuation ring with finite residue field k of odd characteristic. Let G be a general or special linear group or a unitary group defined over O and let $\mathfrak{g}$ denote its Lie algebra. For every positive integer l, let $K^l$ be the l-th principal congruence subgroup of G(O). A continuous irreducible representation of G(O) is called regular of level l if it is trivial on $K^{l+1}$ and its restriction to $K^l/K^{l+1} \simeq \mathfrak{g}(k)$ consists of characters with G(k)-stabiliser of minimal dimension. In this paper we construct the regular characters of G(O), compute their degrees and show that the latter satisfy Ennola duality. We give explicit uniform formulae for the regular part of the representation zeta functions of these groups.

Motivation & Objective

• To construct continuous irreducible regular representations of groups of type A_n over complete discrete valuation rings with finite residue fields of odd characteristic.
• To compute the degrees of these regular representations and establish their uniformity across group types.
• To derive explicit formulae for the regular part of the representation zeta functions of these groups.
• To extend the theory to the special linear and unitary groups by analyzing restrictions to the derived subgroup and Lie algebras.
• To verify Ennola duality in the context of representation zeta functions for these groups.

Proposed method

• Uses the principal congruence subgroups $\mathsf{K}^\ell$ of $\mathbf{G}(\mathfrak{o})$ to define levels of representations and analyze their structure via the Lie algebra $\mathfrak{g}(k)$.
• Constructs regular characters as pullbacks of regular elements in $\mathfrak{g}(k)$, where regularity is defined by the equality of characteristic and minimal polynomials.
• Applies $\mathbf{G}(k)$-equivariant isomorphisms between $\mathfrak{g}(k)$ and its Pontryagin dual $\mathfrak{g}(k)^\vee$ to lift characters to higher congruence levels.
• Employs the adjoint action and orbit structure to classify regular characters by their stabilizers and types $\tau$, using combinatorial data from partition types.
• Establishes surjectivity of reduction maps from $\mathbf{G}'(\mathfrak{o}_m)$ to $\mathbf{G}'(k)$ to ensure compatibility with determinant conditions and $\mathbf{G}'(k)$-orbits.
• Derives the zeta function via summation over orbit types $\tau$, incorporating degrees, orbit counts, and correction factors $u_{\varepsilon_{\mathbf{G}}}^\tau(q)$ and $\iota(\tau, q - \varepsilon_{\mathbf{G}})$.

Experimental results

Research questions

• RQ1 How can regular irreducible representations of $\mathbf{G}(\mathfrak{o})$ for $\mathbf{G} = \mathbf{GL}_n, \mathbf{GU}_n$ be systematically constructed over discrete valuation rings with finite residue fields of odd characteristic?
• RQ2 What is the explicit formula for the degree of a regular representation of level $\ell$ in terms of $n$, $q$, and the orbit type $\tau$?
• RQ3 How do the representation zeta functions of $\mathbf{G}(\mathfrak{o})$ and $\mathbf{G}'(\mathfrak{o})$ relate, and does Ennola duality hold in this setting?
• RQ4 What is the effect of restricting to the special linear or unitary groups on the number and structure of regular characters?
• RQ5 How do the orbit counts and stabilizer sizes in $\mathfrak{g}(k)$ and $\mathfrak{g}'(k)$ affect the enumeration of regular representations?

Key findings

• The degree of a regular character $\chi$ of $\mathbf{G}^\prime(\mathfrak{o}_{\ell+1})$ is given by $\chi(1) = q^{\binom{n}{2}(\ell-1)} \frac{v_{\varepsilon_{\mathbf{G}}}(q)}{u_{\varepsilon_{\mathbf{G}}}^{\tau}(q) \iota(\tau, q - \varepsilon_{\mathbf{G}})}$.
• The number of regular characters of type $\tau$ is $q^{(n-1)(\ell-1)} u_{\varepsilon_{\mathbf{G}}}(q) \frac{\iota(\tau, q - \varepsilon_{\mathbf{G}})}{q - \varepsilon_{\mathbf{G}}} \cdot q^{-1} \prod_d \binom{\sum_e \tau_{d,e}}{\tau_{d,1}, \tau_{d,2}, \ldots} \binom{w_d(q)}{\sum_e \tau_{d,e}}$.
• The regular representation zeta function of $\mathbf{G}^\prime(\mathfrak{o})$ is $\sum_{\ell=1}^\infty \zeta^{\mathrm{reg}}_{\mathbf{G}^\prime(\mathfrak{o}_{\ell+1})}(s)$, with explicit formula involving $\iota(\tau, q - \varepsilon_{\mathbf{G}})^2 / (q - \varepsilon_{\mathbf{G}})$ and degree inverses raised to $s$.
• The number of $\mathbf{G}^\prime(k)$-orbits of type $\tau$ in $\mathfrak{g}^\prime(k)$ is $q^{-1}$ times the number in $\mathfrak{g}(k)$, due to the trace condition and the quotient map $\mathfrak{g}(k) \to \mathfrak{g}^\prime(k)$.
• Ennola duality holds for the zeta functions of $\mathbf{G}(\mathfrak{o})$ and $\mathbf{G}^\prime(\mathfrak{o})$, as evidenced by the symmetric structure in the zeta function formulae.
• The reduction map $\mathbf{G}^\prime(\mathfrak{o}_m) \to \mathbf{G}^\prime(k)$ is surjective on centralizers, ensuring that $\mathbf{G}^\prime(k)$-orbits lift properly from the residue field.