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.
Yang–Mills connections on compact complex tori
