30 P. DEIFT, L. C. LI, AND C. TOMEI
Substitution yields
{*0-,*r,}(A) = (A, [d$lj(A) , d*r,(A))R)
tr Ap[Eji,Esr]
= -«,-,(Ap)r +6rj(Ap)ia .
Thus
{3^, $rs} = 6rj$ia - tts$rj (2.93)
Now consider (Ap)ij = $ij as a function on g£(N,R)* with the Lie-Poisson structure
{F,G}(AP) = tr A
P
[VF T (A
P
) , VG r (A
p
) ] , (2.94)
where VF(AP) = ( —). Then a straightforward computation shows that
u(Ap)ij
{Qij^r^iAp) = 6is$rj - 6rj*is , (2.95)
and we see that, apart from an irrelevant minus sign, the problem reduces to the problem
of computing Poisson brackets on g£(N,R)*.
For 1 r N - 1, let
A
r l
, . . . , A
r r
(2.96)
denote the eigenvalues of the leading submatrix
'(Ap)n ••• (Aph/
(Note that the eigenvalues of the full matrix Ap are already included in the set {Irk} in
(2.89): set z = 1 in (2.87).) The Ar*'s, 1 k r, are invariant,
\rk(g-1Apg) = \rk(Ap) , (2.97)
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