The (cyclic) enhanced nilpotent cone via quiver representations

Abstract

The GL(V )-orbits in the enhanced nilpotent cone V × N (V ) are (essentially) in bijection with the orbits of a certain parabolic P ⊆ GL(V ) (the mirabolic subgroup) in the nilpotent cone N (V ). We give a new parameterization of the orbits in the enhanced nilpotent cone, in terms of representations of the underlying quiver. This parameterization generalizes naturally to the enhanced cyclic nilpotent cone. Our parameterizations are different to the previous ones that have appeared in the literature. Explicit translations between the different parametrizations are given.

Semisimplicity of the category of admissible D-modules

“…We do this for an arbitrary quiver Q with relations I ⊂ CQ. Combining this result with the classification of orbits in the enhanced cyclic nilpotent cone given in [3], we are able to explicitly compute the fundamental group of the orbits in the cone. Recall that a dimension vector d is called sincere if d i = 0 for all i ∈ Q 0 .…”

“…We sketch this parametrization here; the reader is referred to [3] for details. The Gorbits in N ∞ ( , n) correspond to the isomorphism classes of representations of the framed cyclic quiver of dimension v, where the operator a := a −1 • • • • • a 0 acts nilpotent.…”

“…Let P denote the set of all partitions and P the set of all -multi-partitions. In the article [3], we showed that: Theorem 1.1. The G-orbits in the enhanced cyclic nilpotent cone N ∞ ( , n) are naturally labelled by the set Q(n, ) := {(λ; ν) ∈ P × P | res (λ) + sres (ν) = nδ} .…”

“…These orbits were first classified by Johnson [26], extending work of Achar-Henderson [1] and Travkin [38]. In the article [3], we gave a different parametrization of these orbits in terms of the representation theory of the underlying quiver. Let P denote the set of all partitions and P the set of all -multi-partitions.…”

Semi-simplicity of the category of admissible D-modules

“…These orbits were first classified by Johnson [26], extending work of Achar-Henderson [1] and Travkin [38]. In the article [3], we gave a different parametrization of these orbits in terms of the representation theory of the underlying quiver. Let P denote the set of all partitions and P ℓ the set of all ℓ-multi-partitions.…”