Construction of Hyperkähler Metrics for Complex Adjoint Orbits

3.1 The Spectral Curve 24
Using the boundary conditions, we see that
so that
lim A(s, ζ) = 2iAd(g0)(ξ)ζ,
s→−∞
lims→−∞ det(η + A(s, ζ)) = det(η + 2iAd(g0)ξζ)
= det(η + 2iξζ),
and the last expression equals det(η − zζ), as required.
Conversely, let A(s, ζ) be a solution to the Lax-type equation (3.4) with
spectral curve as above. Defining T1, T2, T3 by means of the expression (3.3),
we get a triple that satisfies Nahm’s equations. To check that this solution
obeys the boundary conditions (3.2), we appeal to Biquard’s analysis of solu-
tions to Nahm’s equations in [Biq93]. He shows that any solution defined on
(−∞, 0] has a limit (γ1, γ2, γ3) as s → −∞, with the γi ′ s semisimple and com-
muting with each other. In particular, γ1 + iγ2 is semisimple. On the other
hand, γ1 +iγ2 is nilpotent, since it equals A(−∞, 0), which has characteristic
polynomial η k . It follows that γ1 = γ2 = 0. Now letting s = −∞, ζ = 1, we
see that the elements γ3, −ξ of g have the same characteristic polynomial.
For the Lie algebras we are considering, g = sum, sp m . . .
Example In the case G = SUk, G c = SL(k, C), we fix the maximal
torus t consisting of diagonal matrices, so that z ∈ it is of the form z =
diag(λ1, ..., λk) for real numbers λi. The spectral curve of the adjoint orbit
of ξ = i z assumes the form
2
S : (η − λ1ζ)...(η − λkζ) = 0.
Thus the spectral curve decomposes as k projective lines in T P 1 intersecting
at two common points, namely (η, ζ) = (0, 0) and (0, ∞). The regularity
condition ensures that the λi’s are distinct, so that S does not have multiple
components.