K-Theoretic Donaldson-Thomas Theory of $[{\mathbb{C}}^2\mathrm{/}{\mu }_r]\times \mathbb{C}$ and Factorization

We compute the equivariant K-theoretic Donaldson–Thomas invariants of $[{\mathbb{C}}^2\mathrm{/}{\mu }_r]\times \mathbb{C}$ using factorization and rigidity techniques. For this, we develop a generalization of Okounkov’s factorization technique that applies to Hilbert schemes of points on orbifolds. We show that the (twisted) virtual structure sheaves of Hilbert schemes of points on orbifolds satisfy the desired factorization property. We prove that the generating series of Euler characteristics of such factorizable systems are the plethystic exponential of a simpler generating series. For $[{\mathbb{C}}^2\mathrm{/}{\mu }_r]\times \mathbb{C}$, the computation is then completed by a rigidity argument, involving an equivariant modification of Young’s combinatorial computation of the corresponding numerical Donaldson-Thomas invariants.