Let |=- &1 F be a (G_T_)-invariant pseudo-Kahler form on X_. In this section, we perform symplectic reduction [15] to the right T_-action.
The moment map for this action is denoted 8r: X_Ä t_*
and is called the right moment map. Recall that (t_*)reg$(a_*)regis defined in (1.2).
Proposition 7.1. For all ga # (GG_ss) A_=X_, 8r( ga)=12F $(a) # (t_*)reg. Proof. Since the right T_-action commutes with the G-action, it is clear that 8ris G-invariant. So it suffices to consider 8r(a) for a # A_.
Let v # t_, and let v> and vr denote the infinitesimal vector fields on X_ corresponding to the left and right actions respectively. Since T_A_ is abelian, v>a=vra for all a # A_. Let ; be the real G_T_-invariant 1-form satisfying d;=|. Then
(8r(a), v)=&( ;, vr)a by [1, Theorem 4.2.10]
=&( ;, v>)a
=(8(a), v)
=(12F $(a), v) by Proposition 3.6.
Finally, Theorem 1 says that the image of 12F $ lies in (t_*)reg. Hence we have the proposition. K
Let * # (t_*)reg be in the image of 8r. We consider the reduced space R*=8&1r (*)T_.
Proposition 7.2. Each connected component of 8&1r (*)T_is a copy of the flag domain Y_.
Proof. Since | is pseudo-Kahler, Theorem 1 says that F is non-degenerate. By the inverse function theorem, F $ is a local diffeomorphism.
So there exists a discrete set 1/A_such that (12F $)&1(*)=1. By Proposi-tion 7.1, 8&1r (*)=(GG_ss) 1/(GG_ss) A_. Consequently,
8&1r (*)T_=(GG_) 1. (7.1)
A typical connected component of this space is of the form (GG_) a, a # 1.
Hence we have the proposition. K Consider the inclusion
@: 8&1r (*) Ä X_ (7.2)
and the fibration
\: 8&1r (*) Ä R*. (7.3)
The reduced form |* is defined to be the unique symplectic form on R* such that \*|*=@*|. Since @ and \ commute with the G-action, it is clear that |* is G-invariant. Let
: R*Ä g*
be the moment map of the G-action preserving |*.
By (7.1), write a typical element of R*as ga. If g is the identity coset eG_, we write a= ga for simplicity.
Proposition 7.3. (a)=* # (t_*)reg.
Proof. Pick x # g. By abuse of notation, let x> be the infinitesimal vector field for the G-action on X_, 8&1r (*) or R*, depending on the context. Also, let a denote the appropriate element in any of these three spaces. Since (7.2) and (7.3) commute with the G-action,
@(a)=a, \(a)=a, @
*(x>a)=x>a, \
*(x>a)=x>a.
Since g is semisimple, up to linear combination x=[u, v]. Then ((a), x)=((a), [u, v])=|*(u>, v>)a=\*|*(u>, v>)a=@*|(u>, v>)a
=|(u>, v>)a=(8(a), [u, v])=(*, [u, v])=(*, x). (7.4) So (a)=* and the proposition follows. K
Since Y_is an open set of GCP, it is a complex manifold. Consequently the reduced space R*is complex. Recall that C is defined in (1.7).
Proposition 7.4. The reduced form |* is a G-invariant pseudo-Kahler form on R*. In particular, it is Kahler if and only if * # (t_*)reg& C.
Proof. The G-invariance of |*follows from the discussions in (7.2) and (7.3). So its pseudo-Kahler and Kahler properties remain to be checked.
Consider the elements `i, #i# g from (2.3), indexed by the positive roots :i. Here [`i, #i](:
i, t_){0 can be regarded as a basis of gg_. The almost complex structure inherited from GCP sends `i to #i and #i to &`i. Sub-stituting u=`i and v=#i in (7.4), we get
|*(`>i, #>i)a=|(`>i, #>i)a. (7.5) Since | is pseudo-Kahler, it follows from (7.5) that |* is pseudo-Kahler too.
In fact, |* is Kahler if and only if (7.5) is positive for all (:i, t_){0.
Following the argument in (7.4), we see from (2.5) that
|*(`>i, #>i)a=\(*, :i). (7.6) Here the sign \ is positive when :i is compact and negative when :i is non-compact. So (7.6) is positive for all (:i, t_){0 if and only if
* # (t_*)reg& C, and this is the equivalent condition for |*to be Kahler. K For i=1, 2, consider the reduced spaces (R*i, (|i)*i), with moment maps
i: R*iÄ g*. By the previous proposition, these reduced spaces are pseudo-Kahler. So we can compare them under the notions of t and = introduced in (1.12).
Proposition 7.5. Suppose that R*
i have the same number of connected components. Then (|1)*1t(|2)*2if and only if *1t*2, and (|1)*1=(|2)*2 if and only if *1=*2.
Proof. Suppose that this proposition has been proved for all connected reduced spaces. Let R* be a reduced space, possibly non-connected. For i=1, 2, let Y_aibe connected components of R*. By Proposition 7.3, their
moment maps satisfy i(ai)=*. So by the present proposition for connected reduced spaces, Y_a1and Y_a2are isomorphic pseudo-Kahler manifolds. We conclude that all connected components of R*are isomorphic to one another, and so the present proposition holds for non-connected reduced spaces too.
From this observation, we only have to prove the proposition for connected reduced spaces. So assume that R*i are connected for i=1, 2. Write R*i=(GG_) ai for some ai# A_.
Suppose that *1t*2. Thus there is a coadjoint orbit O/g* which contains *1 and *2. By Proposition 7.3, i(ai)=*i. By Theorem 1 and Proposition 7.1, *i# (t_*)reg/t*, so the isotropy subgroup of *iin G is G_. Hence O=GG_. So iis a diffeomorphism from (GG_) aionto the elliptic orbit O. In fact, since i is G-equivariant, it identifies (|i)*
i with the KirillovKostant symplectic form |KK on O. We conclude that (|1)*1t
|KKt(|2)*
2.
Conversely, if (|1)*1t(|2)*2, then i have the same image O. By Proposition 7.3, i(ai)=*i# O, so *1t*2.
The last part of this proposition remains to be proved, where t is replaced with =. Suppose that *1=*2. By (7.4), for all u, v # g,
(|1)*1(u>, v>)a1=(*i, [u, v])=(|2)*2(u>, v>)a2. (7.7) Consider the G-equivariant biholomorphic map
}: Y_a1Ä Y_a2, }( ga1)= ga2. (7.8) By (7.7), }*(|2)*2 and (|1)*1 agree on a1. By G-invariance, they agree everywhere. So } preserves the pseudo-Kahler structures and (|1)*1=(|2)*2. Conversely, suppose that *1{*2. If *iare in different coadjoint G-orbits, then the first part of the proposition says that (|i)*iare not symplectomorphic, so in particular (|1)*
1{(|2)*
2. Hence we may assume that *i are in the same orbit. Each connected component of (t_*)reg/g* intersects a G-orbit at most once. From *i# (t_*)reg, *1{*2and *1t*2we conclude that *iare in different connected components of (t*)_ reg. The holomorphic map (7.8) fails to preserve the pseudo-Kahler structures because (7.6) shows that there is a sign problem arising from (2.5). Other symplectomorphisms between (|i)*i have to permute the connected components of (t*)_ reg, so they cannot be holomorphic. We conclude that (|1)*1{(|2)*2. This proves the proposition. K
Proof of Theorem 4. The theorem follows directly from Propositions 7.1 through 7.5. K