An Application of Spherical Geometry to Hyperkähler Slices. (24th June 2021)

• Record Type: Journal Article
• Authors: Crooks, Peter; van Pruijssen, Maarten
• Is Part Of: Canadian journal of mathematics. Volume 73:Number 3(2021)
• Page Start: 687
• Page End: 716
• Publication Date: 2021-06-24
• Subjects: 20G20 -- 53C26 -- 14M17; hyperkähler quotient -- Slodowy slice -- spherical geometry; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/S0008414X20000127
• Languages: English
• ISSNs: 0008-414X

Abstract

Abstract: This work is concerned with Bielawski's hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice with the data of a complex semisimple Lie group $G$, a reductive subgroup $H\subseteq G$, and a Slodowy slice $S\subseteq \mathfrak{g}:=\text{Lie}(G)$, defining it to be the hyperkähler quotient of $T^{\ast }(G/H)\times (G\times S)$ by a maximal compact subgroup of $G$ . This hyperkähler slice is empty in some of the most elementary cases ( e.g., when $S$ is regular and $(G, H)=(\text{SL}_{n+1}, \text{GL}_{n})$, $n\geqslant 3$ ), prompting us to seek necessary and sufficient conditions for non-emptiness. We give a spherical-geometric characterization of the non-empty hyperkähler slices that arise when $S=S_{\text{reg}}$ is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called $\mathfrak{a}$ - regularity of $(G, H)$ . This $\mathfrak{a}$ -regularity condition is formulated in several equivalent ways, one being a concrete condition on the rank and complexity of $G/H$ . We also provide a classification of the $\mathfrak{a}$ -regular pairs $(G, H)$ in which $H$ is a reductive spherical subgroup. Our arguments make essential use of Knop's results on moment map images and Losev's algorithm for computing Cartan spaces.