An Application of Spherical Geometry to Hyperkähler Slices. (24th June 2021) - Crooks, Peter; van Pruijssen, Maarten
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 HG, and a Slodowy slice Sg:=Lie(G), defining it to be the hyperkähler quotient of T^* (G/H)× (G× 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)=(SL_n+1, GL_n), n3 ), 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_reg is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called a - regularity of (G, H) . This 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 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.