AN APPLICATION OF SPHERICAL GEOMETRY TO HYPERKÄHLER SLICES
Canadian Journal of Mathematics - Journal Canadien de Mathematiques
This work is concerned with Bielawski's hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice to the data of a complex semisimple Lie group G, a reductive subgroup H ⊆ G, and a Slodowy slice S ⊆ g := 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) = (SLn+1, GLn), n ≥ 3),
... n), n ≥ 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 = Sreg 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. Contents 2010 Mathematics Subject Classification. 20G20 (primary); 53C26, 14M17 (secondary).