University of Sydney Algebra Seminar

Anthony Henderson (University of Sydney)

Friday 14th March, 12:05-12:55pm, Place: TBA

Diagram automorphisms of quiver varieties

(Joint work with Anthony Licata (ANU).) The McKay correspondence is a bijection between (isomorphism classes of) finite subgroups $\Gamma$ of $SL_2(\mathbb{C})$ and simply-laced simple Lie algebras $\mathfrak{g}$, arising because both are parametrized by the Dynkin diagrams of type A/D/E. Non-simply-laced simple Lie algebras can be brought into this correspondence also: they arise as the fixed-point subalgebras of simply-laced simple Lie algebras under automorphisms arising from automorphisms of the Dynkin diagrams, and the latter come from embeddings of one finite subgroup of $SL_2(\mathbb{C})$ in another. It is interesting to investigate how deep the correspondence goes, that is, what other connections there are between objects defined in terms of $\Gamma$ and objects defined in terms of $\mathfrak{g}$. We study the existence of isomorphisms between Nakajima quiver varieties of finite type (these can be thought of as attached to $\Gamma$) and Slodowy varieties (attached to $\mathfrak{g}$). In type A there is a theorem of Maffei along these lines; we give a partial generalization to other types by describing the fixed-point subvarieties of diagram automorphisms.