[Paper Review] A McKay Correspondence in Positive Characteristic

Christian Liedtke|arXiv (Cornell University)|Jul 13, 2022

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes a McKay correspondence for finite and linearly reductive subgroup schemes of SL₂ in positive characteristic p ≥ 7, extending the classical McKay correspondence beyond the tame case. It defines conjugacy classes for finite group schemes via Hopf algebra representations and proves a bijection between conjugacy classes, irreducible representations, and vertices of affine Dynkin diagrams, enabling a McKay correspondence for all rational double point singularities in this characteristic range.

ABSTRACT

We establish a McKay correspondence for finite and linearly reductive subgroup schemes of $\mathrm{SL}_2$ in positive characteristic. As an application, we obtain a McKay correspondence for all rational double point singularities in characteristic $p\geq7$. We discuss linearly reductive quotient singularities and canonical lifts over the ring of Witt vectors. In dimension 2, we establish simultaneous resolutions of singularities of these canonical lifts via $G$-Hilbert schemes. In the appendix, we discuss several approaches towards the notion of conjugacy classes for finite group schemes: This is an ingredient in McKay correspondences, but also of independent interest.

Motivation & Objective

• To extend the classical McKay correspondence to positive characteristic, particularly in the wild case where p divides the group order.
• To define a meaningful notion of conjugacy classes for finite group schemes over algebraically closed fields of positive characteristic.
• To establish a McKay correspondence for rational double point singularities in characteristic p ≥ 7 using linearly reductive group schemes.
• To explore the structure of derived categories and canonical lifts over Witt vectors for these singularities.
• To compare and evaluate multiple approaches to conjugacy classes in finite group schemes, especially via Hopf algebra representations.

Proposed method

• Uses the category of finite-dimensional, k-linear representations of a finite group scheme G to define linearly reductive group schemes via semi-simplicity.
• Applies the extended adjoint representation on the Hopf algebra H⁰(G, O_G)* to define conjugacy classes via simple subrepresentations.
• Constructs the McKay graph Γ and its affine extension  Γ̂ using isomorphism classes of irreducible representations and conjugacy classes.
• Employs G-Hilbert schemes to achieve simultaneous resolutions of canonical lifts of quotient singularities over the ring of Witt vectors.
• Analyzes the structure of Hopf algebras associated to group schemes like μ_p^n and α_p, particularly their adjoint and extended adjoint representations.
• Compares six approaches to conjugacy classes in finite group schemes, favoring the extended adjoint representation for linearly reductive cases.

Experimental results

Research questions

• RQ1 Can a McKay correspondence be established for rational double point singularities in positive characteristic p ≥ 7, even when the group order is divisible by p?
• RQ2 What is an appropriate generalization of conjugacy classes for finite group schemes in positive characteristic, especially when the group is not étale?
• RQ3 How do the representations of the Hopf algebra H⁰(G, O_G)* relate to conjugacy classes and irreducible representations in the linearly reductive setting?
• RQ4 To what extent can canonical lifts of quotient singularities over the ring of Witt vectors admit simultaneous resolutions via G-Hilbert schemes?
• RQ5 Which of several proposed definitions of conjugacy classes for finite group schemes yields a consistent and useful McKay correspondence?

Key findings

• A McKay correspondence is established for all rational double point singularities in characteristic p ≥ 7 using finite and linearly reductive subgroup schemes of SL₂.
• The number of conjugacy classes of a finite and linearly reductive group scheme G over k is equal to the number of isomorphism classes of simple representations of G.
• The extended adjoint representation on the Hopf algebra H⁰(G, O_G)* decomposes into isotypical components that correspond to conjugacy classes, providing a representation-theoretic definition of conjugacy.
• For linearly reductive group schemes, the dual of the regular representation of G_abs = G(k) decomposes into components indexed by conjugacy classes, suggesting a duality between conjugacy classes and irreducible representations.
• Canonical lifts of linearly reductive quotient singularities over the ring of Witt vectors admit simultaneous resolutions via G-Hilbert schemes in dimension 2.
• The extended adjoint representation approach yields a consistent definition of conjugacy classes even in the wild case, unlike the adjoint representation on A or A*, which may not be semi-simple.