Construction of Hyperkähler Metrics for Complex Adjoint Orbits

3.2 Regularity 26
g c belongs to a single adjoint orbit, say Oβ. In order to prove the lemma,
we must show that Oβ is a regular orbit. In fact we shall see next that the
boundary value A(0) = −Φ also belongs to this orbit. Since A(s) → −Φ as
s → 0, it follows that −Φ(ζ) ∈ Oβ, where Oβ denotes the closure of Oβ in
g c . Therefore the same is true of the G c −adjoint orbit of −Φ(ζ):
O−Φ(ζ) ⊆ Oβ. (3.5)
Now O−Φ(ζ) is by assumption a regular orbit. Since regular orbits have
maximal dimension, they are not contained in the closure of any other orbit.
Therefore it follows from (3.5) that we must have O−Φ(ζ) = Oβ. //
The next result, whose proof will be postponed to the next chapter, shows
that there are plenty of regular solutions A(s, ζ) defined on (−∞, 0]. Denote
by M(ξ)reg the subset of M(ξ) consisting of those triples (T1, T2, T3) for which
the associated A(s, ζ) is regular.
Proposition 3.3 M(ξ)reg is an open dense subset of M(ξ).
For simplicity, we shall assume for the rest of this chapter that G =
SUk, G c = SL(k, C).
Now as observed in Chapter 2, if A(s, ζ) is regular we obtain a flow of
line bundles on the spectral curve S by considering the eigenvector bundles
ker(η + A(s, ζ) T ). This is a linear flow on Jac(S); more precisely, the flow
(ker(η + A(s, ζ) T )) ∗ is of the form ML s , where M has degree k(k − 1) and
L s is the restriction to S of the line bundle on T P 1 defined by the transition
function exp(sη/ζ) with respect to the usual open covering of T P 1 . We shall
consider next the inversion of this procedure, following the lines of [Hit83],
as summarized in Chapter 2.
Remark Roughly speaking, through this procedure one may identify
M(ξ)reg/SUk with an open subset of a “real Jacobian” Jac(S) R - observe