Special McKay Correspondence

• Special McKay correspondence is a categorical refinement that links the geometry of minimal resolutions of surface singularities with special representations defined by Wunram.
• It employs semiorthogonal decompositions and root-stack constructions to align derived categories with exceptional divisors and non-special representations.
• Extensions of the correspondence utilize GIT, Floer theory, and quiver methods to generalize classical results to positive characteristic and higher dimensions.

The special McKay correspondence is a categorical and representation-theoretic refinement of the classical McKay correspondence, extending from the setting of finite subgroups of $\mathrm{SL}_2$ to more general finite subgroups of $\mathrm{GL}_2$, as well as to higher dimensions, positive characteristic, and singular settings. Central to this correspondence is the relationship between the geometry of the minimal resolution of quotient surface singularities, the derived category of coherent sheaves on quotient stacks, and the modular structure inherent in their representation theory. In many cases, the "special" label refers to a distinction—rooted in Wunram's notion—between certain irreducible representations (called special) and the remainder (non-special), the former precisely matching the exceptional loci in a minimal (or maximal) resolution.

1. Formulation: Semiorthogonal Decomposition and Special Representations

Let $G\subset \mathrm{GL}_2(k)$ be a finite small subgroup acting linearly on $A^2 = \operatorname{Spec} k[x,y]$ with $k$ algebraically closed and $\operatorname{char} k = 0$. The quotient stack $[A^2/G]$ has a derived category $D^b([A^2/G])$ equivalent to $D^b(\mathrm{coh}\,(R\rtimes G))$, where $R=k[x,y]$. The minimal resolution $\tau:Y\to X=A^2/G$ enables the construction of an integral functor $\Phi: D^b(\mathrm{coh}\,Y)\to D^b([A^2/G])$ which is fully faithful but not, in general, an equivalence unless $G\subset \mathrm{SL}_2$ (the crepant case).

The special McKay correspondence (Ishii et al., 2011) asserts the existence of a semiorthogonal decomposition: $$D^b([A^2/G]) = \langle E_1,\ldots, E_n,\, \Phi D^b(\mathrm{coh}\,Y)\rangle$$ where:

• $n$ equals the number of irreducible non-special representations of $G$ (per Wunram's criterion),
• $\{E_i\}$ forms a full exceptional collection generating the right orthogonal to $\mathrm{Im}\,\Phi$,
• $\mathrm{Im}\,\Phi$ is generated by the special representations, which biject onto the irreducible components of the exceptional locus in $Y$.

In the abelian case (e.g., cyclic $G$), the special representations are specified explicitly via continued fraction invariants, and the exceptional collection $\{E_d\}$ is constructed as equivariant structure sheaves on certain fat points, twisted by characters reflecting the combinatorics of Wunram special and non-special indices.

2. Explicit Constructions: Abelian Case, Root Stacks, and Gluing

For a cyclic group $G \cong \langle 1/n(1,q)\rangle$, special representations correspond to distinguished characters determined by the continued fraction expansion $n/q = b_1 - 1/(b_2 - \ldots)$. The exceptional objects $E_d$ for non-special $d$ are constructed as: $E_d := R_{j_t} \otimes \rho_{d - (j_t-1)q}$ where $R_{j_t} = k[x,y]/(x,y^{j_t})$ and $\rho_{d-(j_t-1)q}$ is a character of $G$. These objects form a semiorthogonal exceptional collection, with each $E_d$ corresponding to a unique non-special character, and with their socles at the origin realizing the non-special simple modules.

In the non-abelian case, the strategy is to reduce to root stacks: given intermediate stacks with residual stabilizers along divisors, one removes these by constructing successive rth root stacks, iteratively applying semiorthogonal decompositions tied to the geometry of root constructions. The final gluing step is to patch these exceptional sequences onto $D^b(\mathrm{coh}\,Y)$, using Orlov's blowup formula for further (−1)–curves, yielding the global semiorthogonal decomposition (Ishii et al., 2011).

3. The Correspondence: Dihedral and Non-Abelian Settings

For dihedral groups $D_{2n} \subset \mathrm{GL}_2(\mathbb{C})$, recent work establishes the special McKay correspondence via explicit GIT (moduli of $G$-constellations), demonstrating that every resolution $Y \to \mathbb{C}^2/G$ dominated by the maximal resolution arises as the fine moduli space $M_\theta$ for some generic $\theta$ (Capellan, 2024). The tautological bundles $R_\rho$ on $Y$ are constructed from the universal family, and their Chern classes reflect the full representation-theoretic data. The locus "socle-top" correspondence between exceptional curves and irreducible representations is realized by analyzing the fibers of the universal family and the derived equivalence induced by the Fourier–Mukai transform.

A key role is played by the canonical stack and root constructions that modify stabilizer structures during the transition from $G\cap \mathrm{SL}_2$ to $G$ and produce appropriate exceptional summands in the derived category, ultimately aligning with the non-special part of the McKay quiver.

4. Extensions: Positive Characteristic, Quiver, and Floer-Theoretic Perspectives

In positive characteristic, the special McKay correspondence remains valid for linearly reductive subgroup schemes of $\mathrm{SL}_2$ (Liedtke, 2022). Here, the minimal resolution coincides with the $G$-Hilbert scheme, and the bijection is realized among irreducible $G$-representations, components of the exceptional locus, and conjugacy classes. The McKay graph in this context is identified with affine Dynkin diagrams, with the "special" bijection surviving under canonical lifting to $W(k)$.

The quiver structure underlying the reconstruction algebra is controlled by the data of minimal $G$-generators and socle filtrations of the $G$-cluster structure sheaves, as elaborated in (Ishii et al., 2016). This enables one to reconstruct the path algebra and relations corresponding to the derived category of the minimal resolution from the representation side.

Analytically, symplectic and Floer-theoretic methods confirm the correspondence via age grading and Conley–Zehnder indices: the rank of the positive symplectic cohomology of $Y$ equals the number of conjugacy classes of $G$, and the degree-2 part (corresponding to age $=1$) matches the number of exceptional divisors in the minimal resolution (McLean et al., 2018).

5. Higher and Singular Dimensional Generalizations

The special McKay correspondence extends into dimension three and to certain non-isolated quotient singularities, as in the context of non-abelian finite subgroups of $\mathrm{SL}_3(\mathbb{C})$ (Ebeling, 2017). For these, the algebra of invariants is investigated through the lens of Poincaré series, and a direct relation is established between the algebraic data (the ring of invariants and its relations) and characteristic polynomials of Coxeter elements associated with hyperplane sections (often Kleinian surface singularities). The rational form of the Poincaré series reflects Coxeter-theoretic data, and its relation with affine and non-affine Coxeter polynomials represents an incarnation of the special McKay correspondence at the level of generating series for invariants.

6. Categorical, Elliptic, and Geometric Refinements

The correspondence is further enriched in the setting of derived categories, where it often takes the form of a fully faithful embedding of $D^b(\mathrm{coh}\,Y)$ into $D^b([A^2/G])$, with a complementary exceptional collection mirroring the non-special part of McKay's quiver (Ishii et al., 2011). This framework supports semiorthogonal decompositions, tilting bundles corresponding to tautological representations, and explicit quiver constructions for the endomorphism algebras, as realized even in non-toric cases such as $D_4$ via VGIT (Abdelgadir et al., 2024).

Refinements arise in equivariant and elliptic cohomology: elliptic genera and their localization formulas generate nontrivial theta function identities, whose functional equations reflect the equivalence of contributions coming from the resolution and the orbifold presentations, particularly in the symplectic case. Here, the "elliptic special McKay correspondence" articulates the equality of equivariant elliptic classes, refining previous numerical and cohomological correspondences (Mikosz et al., 2019).

7. Tabular Summary: Key Features of the Special McKay Correspondence

Aspect	Description	Sources
Ambient Group	$G \subset \mathrm{GL}_2(k)$ (and extensions to $\mathrm{SL}_n$, $n\ge 2$)	(Ishii et al., 2011, Ebeling, 2017)
Special Representations	Wunram's definition: those corresponding bijectively to exceptional divisors	(Ishii et al., 2011, Ishii et al., 2016)
Semiorthogonal Decomposition	$$D^b([A^2/G]) = \langle E_1,\ldots,E_n,\,\Phi D^b(\mathrm{coh}\,Y)\rangle$$	(Ishii et al., 2011)
Gluing Construction	Root-stack methods to assemble exceptional collections in non-abelian/non-crepant settings	(Ishii et al., 2011)
Positive Characteristic	Correspondence holds for linearly reductive $G$; canonical lifts and $G$-Hilb yield simultaneous resolutions	(Liedtke, 2022)
Floer-Theoretic Version	Positive symplectic cohomology $SH_+(Y)$ graded by age, special part matches exceptional divisors	(McLean et al., 2018)
Elliptic Refinement	Equivariant elliptic genera produce theta identities reflecting the correspondence	(Mikosz et al., 2019)
Higher-Dimensional Case	Relation between Poincaré series of invariants and Coxeter (monodromy) polynomials of hyperplane sections	(Ebeling, 2017)

The special McKay correspondence thus unifies the geometry of surface singularities, representation theory of finite groups, derived and categorical structures, and—in advanced settings—the moduli of sheaves and intersection-theoretic invariants, providing a robust framework that extends and refines the classical correspondences in algebraic and symplectic geometry.