Pseudokähler forms on complex Lie groups

LetG be a semisimple complex group with real formGR. We dene and study a pseudokahler form that is dened on a neighbhorhood of the identity in G and is invariant under left and right translation by GR.

down the pseudokähler form on M to the open set f (M ) ⊂ G.The new form turns out to be closely related to the form Z, W → κ(Z, W ). Our main results are as follows: Let G R ×G R act on ig R ×G R by the rule (r1,r2) (iX, r) = (iAd r1 X, r 1 rr −1 2 ); then f equivariant.Define M := {(iX, r) : df is nonsingular at (iX, r)} ⊂ ig R × G R .This makes M a complex manifold, with complex structure J induced from the complex structure on G.A useful description of M is: ∈ M if and only if ad X has an eigenvalue of nπ for some nonzero integer n.Equivalently, for p := e iX , (iX, r) / ∈ M exactly when either Ad p has an eigenvalue of −1 or Ad p fixes a vector in g not fixed by ad X .(Proof in §2.) Regard ig R × G R as the cotangent bundle of G R .As such, there is a canonical real 1-form λ on ig R × G R such that ω := dλ is an (exact) nondegenerate symplectic form.On the other hand, let φ : ig R × G R → R, (iX, r) → κ(X, X), which is a G R × G R -invariant function.These objects are related: Theorem 2. On the complex manifold M ⊂ ig R × G R , we have 2λ = d c φ. (Proof in §4.) As an immediate corollary, we have: We seek to compute the pseudokähler form in terms of a reasonable collection of vector fields on M .Let Z ∈ g, that is to say, a tangent vector to G at the identity 1.As usual, we may identify Z with a left G-invariant vector field on G. Let Z denote the vector field on M obtained by pulling back the left G-invariant vector field Z on G via the map f .These vector fields, which we call canonical vector fields, are the ones we shall use throughout for computations.We also need to define several linear transformations on g.Let (iX, r) ∈ M and write p := e iX .First, we define A p : g → g, . (This makes sense, by Theorem 1.) We also define F iX : g → g by F iX (Z) := d ds s=0 log(pe sZ ).(Here log denotes a local inverse for exp, returning a neighborhood of p in G to a neighborhood of iX in g.) Finally, define E iX := F iX • A p • Ad p .Our result is: Theorem 4. For Z, W ∈ g and (iX, r) ∈ M , we have that Z, W (iX,r) = κ(E iAd −1 r X Z, W ). (Proof in §5.) Theorem 4 is useful since E iX is easy to understand: if ad X is diagonalizable, then E iX is also diagonalizable, has the same eigenspaces as ad X , and Documenta Mathematica 5 (2000) 595-611 its eigenvalues can be expressed in terms of the corresponding eigenvalues of ad X .In particular, if X lies in a Cartan subalgebra t R of g R , then one has a simple expression for , at (iX, 1) when expressed using canonical vector fields corresponding to elements of g that are root vectors or vectors in t.We refer the reader to §6 for the precise statement.Additionally, Theorem 5.The signature of the pseudokähler form is constant on M , and is equal to the signature of the Hermitian form Z, W → κ(Z, W ) on g. (Proof in §6.) We produce such a set M : Theorem 6.Let ψ : G → GL(V ), g → gl(V ) be a finite-dimensional representation that is defined over R and is faithful modulo the center of G (e.g. the adjoint representation).Define M ⊂ ig R × G R , where (iX, r) ∈ M if and only if for each eigenvalue λ of ψ(X), |Re λ| < π/2.Then In some applications, it is more convenient to replace the above canonical vector fields with tangent vectors that are either tangent ("orbital vectors") or transverse ("vertical vectors") to the G R × G R -orbits.We set up the notation and compute the pseudokähler form using these vectors ( §7).The imaginary part of the pseudokähler form is particularly easy, and from it, one easily computes the moment map: Theorem 7. Relative to the symplectic form ω, the moment map µ The present paper extends recent results of Gregor Fels.In [F], pseudokähler forms are defined on certain complex domains which turn out to be subsets of M .In §7, we verify that the restriction of , to these domains coincides with Fels' definition.That paper uses orbital and vertical vector fields, and includes discussions of the moment map and CR-structures.I am grateful to G. F. for sharing a copy of his preprint with me.I also thank Alan Huckleberry for suggesting this problem to me and for helpful conversations.§1.The Group Action Let G be a connected complex semisimple algebraic group, endowed with a complex conjugation g → g, defining a real form G R ⊂ G (the fixed point subgroup of the complex conjugation).The real group G R × G R acts on G by the rule (r1,r2) With this action, the map f We shall often be particularly interested in the case when X is semisimple, and we now set up some notation.If X is semisimple, then it is contained in t R , where t is a (complex) Cartan subalgebra of g that is stable under complex conjugation.Also p := e ix ∈ T , where T is the maximal (complex) R-torus of G with Lie (T ) = t.We have a root system Φ(T, G) consisting of characters α : T → C * , with differentials dα : t → C. (By abuse of notation, we write −α for the inverse of α.) Roots are real, imaginary, or complex according to whether α = α, −α, or neither.Imaginary roots arise in two ways, according to whether the set of real points of the corresponding root sl(2) is isomorphic to sl(2, R) or su(2), and are respectively "noncompact imaginary" or "compact imaginary" roots.We have that dα(iX) ∈ iR (resp.R) if α is real (resp.imaginary) and hence α(p) = e dα(iX) ∈ U (1) (resp.R >0 ).We recall some related facts (see [BF]).Let In this section, we define the "canonical vector fields," which are global vector fields on a dense open subset M ⊂ ig R ×G R that are associated to elements of g.We define and provide a characterization of M (2.1).Given a point (iX, r) ∈ M and Z ∈ g, we produce a curve through (iX, r) in M whose tangent vector at (iX, r) is the canonical tangent vector associated to Z (2.6).
We would like to define global vector fields on ig R × G R by pulling back the left invariant vector fields on G via the map f ; that is, given Z ∈ g, we would like to define a vector field This works precisely at the points where df is an isomorphism.We define M = {(iX, r) : ad X has no eigenvalue of πn for any nonzero integer n} ⊂ ig R × G R .Letting p = e iX , we see that (iX, r) fails to be in M exactly when either Ad p has an eigenvalue of −1, or Ad p fixes a point in g not fixed by ad X .Below we prove: Theorem 2.1.The differential of f is an isomorphism at (iX, r) precisely when (iX, r) ∈ M .
Thus an element Z ∈ g yields a globally-defined nonvanishing vector field Z on M , which we call the canonical vector field associated to Z.We see, by taking a basis of g, that the tangent bundle of M is trivial.If we denote the complex structure on M by J, we have that J( Z) = (iZ) ∧ , where i is the complex structure on the vector space g.The action of G R × G R on M induces an action on vector fields, and For df to be nonsingular at (iX, r), we need the exponential map exp : ig R → B(G) to be nonsingular at iX, and we need the multiplication map B(G) × G R → G to be nonsingular at (e iX , r).Thus 2.1 follows from 2.2 and 2.3 below.
and is an isomorphism exactly when ad Y has no eigenvalue of 2πin, n ∈ Z \ {0}.
(In particular, if Y ∈ g has no eigenvalue of 2πin (n a nonzero integer), then there is a well-defined ) is an isomorphism if and only if Ad p has no eigenvalue of −1.
Before proving 2.3, we need to define an important linear operator on g.Let X ∈ g R and let p = e iX .Assume that Ad p has no eigenvalue of −1.We define . We will often use the following properties of A p : Lemma 2.4. (1) the Ad-invariance of κ.To prove (3), let Z = A p Z and W = A p W . Then the left side of (3) is κ( ).The proofs of ( 4), ( 5), and ( 6) are similar, using also that A p and Ad p fix X.
Proof.Since Z(G) is smooth, any tangent vector at p can be written as dl p Z for some Z ∈ g.If Z = −Ad p Z then the curve pe tZ is contained in Z(G) since pe tZ = p −1 e tZ and (pe tZ ) −1 = e −tZ p −1 = p −1 e −tAdpZ = p −1 e tZ .Hence all such Z give tangent vectors in Z(G).Note that since p = p −1 , Z → Ad p Z gives a complex conjugation on the vector space g; the choice of Z above amounts to the pure imaginary elements of g for this real structure.The lemma follows since dim R (Z(G)) = dim C g. Let (iX, r) ∈ M and Z ∈ g.Since t → e iX re tZ gives an integral curve (starting at e iX r) for the left invariant vector field associated to Z, we can obtain (for t small) an integral curve at (iX, r) for Z, by locally inverting f .The resulting curve δ is described below.Unfortunately this curve is unwieldy for computations.Instead, we produce a simpler curve γ in ig R × G R which has tangent vector Z at (iX, r) but not at other points on the curve.(This will be sufficient for applications.) Proposition 2.6.Let (iX, r) ∈ M , with p = e iX , and let Z ∈ g.Define the following curves in M : , and δ is an integral curve for Z starting at (iX, r).Proof.(a) Both curves have a value of (iX, r) at t = 0.
(b) We must verify that the curves actually lie in ig R × G R .For γ, one can use 2.5 to show that pe tAp•AdriIm Z ∈ Z(G), and using 2.4(1), it is easy to verify that Ad r Z−A p •Ad r iIm Z ∈ g R .For δ, one has that pre tZ e −tZ r −1 p ∈ Z(G) since its Documenta Mathematica 5 (2000) 595-611 complex conjugate equals its inverse.The fact that p(t) −1 pre tZ ∈ G R turns out to be equivalent to p(t) 2 = pre tZ e −tZ r −1 p, which is true by definition.
(c) Proving that the curves have the correct derivatives follows from pushing them forward via f .We have that f (γ(t)) = pe tAp•AdriIm Z e t(Adr −Ap•Adr iIm Z) r, whose derivative at t = 0 is dl p • dr r • Ad r Z = dl pr Z, as required.Note that at other values of t, tangent vectors for this curve do not coincide with the left invariant vector field on G! However, we do have f (δ(t)) = pre tZ , as required.§3.The Differential of the Logarithm Map Throughout this section, let (iX, r) ∈ M and let p = e iX .We define: This section is devoted to listing properties of this map.By definition F iX = d(log iX •l p ), with the differential taken at the identity.
Near the identity element of G, the map l p −1 • exp • log iX •l p is (defined and) the identity function, so after taking differentials at the identity, we have Proof.
Let T = −ad iX and λ ∈ C. We must prove that We compute that Proof.
(1,2) are easy.By substitution, (3 ) by the associativity of the Killing form.Then (4) follows from (1) and (3).For (5), we have • ad iX = I • ad iX .§4.The Liouville Form on M and its Exterior Derivative We recall that the cotangent bundle to any real manifold possesses a canonical 1-form λ and that ω := dλ is a nondegenerate exact symplectic form (see [A], [CG]).We have identified We wish to obtain a formula for λ( Z); this conveniently expresses the restriction of λ to M .By 2.6, we have that λ( Z) (iX,r) = κ(X, d dt t=0 e t(Adr Z−Ap•Adr iIm Z) ) = κ(X, Ad r Z − A p • Ad r iIm Z), where p = e iX .This can be sharpened: Proof.We must show that κ(X, Ad r iIm Z) = κ(X, A p •Ad r iIm Z).This follows from 2.4(3), since Ad p • A p (X) = X.
On M , we have a complex structure J.Even prior to the explicit computation of dλ, we have the following important observation: This is a consequence of 4.4, for which we recall the customary notation.On any complex manifold, the exterior derivative is written as the sum From d 2 = 0, we know that ∂∂ = −∂∂, and hence dd c = 2i∂∂ = −d c d.One can show (e.g. using local coordinates) that if φ is a smooth function and X a vector field on M , then d c φ(X) = dφ(JX); by the derivation property, if µ is a 1-form, then
We return to the computation of dλ.First, Similarly, we have Summing the terms and using 2.4(1), we find We have proved: , real-valued 2-form on M , and coincides with the restriction to M of the standard (cotangent bundle) symplectic form on T * (G R ).
Recall that on any complex manifold, a closed, nondegenerate, real 2-form ω for which the complex structure is an isometry yields a pseudokähler form, by the rule Z, W = ω(J Z, W ) + iω( Z, W ).Here Z, W → ω(J Z, W ), the real part of Z, W , is a real, J-invariant, symmetric bilinear form (which need not be positive definite), and the imaginary part of , is just ω.
In our situation, we have: Theorem 5.3.The pseudokähler form associated to ω is Note that 5.1(2) shows independently that , is Hermitian.§6.Evaluation of the Pseudokähler Form on a Basis Our next goal is to compute , with respect to a natural basis of vector fields at (iX, r), in the (generic) case that X is semisimple.Without loss of generality, we assume that r is the identity element of G R .We use notation involving T and t as in §1.
It is clear that E iX preserves t and each root space g α .Hence for our Hermitian form, t ⊥ g α and g α ⊥ g β unless β = −α.So, the only products we need compute are Z, W for Z, W ∈ t, and Z α , Z −α , where Z α ∈ g α .
By 2.4(1) and the definition of E iX , it follows easily that: Lemma 6.2.For Z, W ∈ t, we have From this, it is easy to describe the signature of , on t: suppose the connected component of 1 in T R is a product of n circles and m real lines (here n + m is the complex dimension of T ).Then , is negative-definite on the complexified Lie algebra of the circles and positive-definite on the lines, and these two subspaces of t are perpendicular.For: in computing signatures, we may assume that Z ∈ t R .If Z ∈ t R , then in the former case dα(Z) ∈ iR, and in the latter, dα(Z) ∈ R. Also for Z ∈ t R , Z, Z = α∈Φ(T,G) (dα(Z)) 2 .Also if Z, W ∈ t R but are of "opposite types," the last lemma shows that Recall that by our definition of M , we have α(p) = −1.Also, either α(p) = 1, or dα(iX) = 0 and α(p) = 1.
This shows that if α is not imaginary, then , is isotropic on g α ⊕ g −α .Suppose that α is imaginary; we wish to see whether , is positive-or negativedefinite on g α .In the copy of sl(2) (sl(α) ⊂ g) corresponding to α, recall that we may pick a basis By explicit computation, one sees that one can pick In the latter case, since κ(Z α , Z −α ) > 0 we Documenta Mathematica 5 (2000) 595-611 already see that , is positive on g α if α is noncompact and negative if α is compact.In the former case, we get the same information: since α is imaginary, we have dα(iX) ∈ R, and α(p) = e dα(iX) > 0. Note then that dα(iX)/(1 − α(p)) < 0; it then follows that the sign of Z α , Z α is .Summarizing the above, we have: Theorem 6.4.Suppose that (iX, 1) ∈ M and e iX ∈ T for some maximal Rtorus T of G. Write T = T s • T a , the decomposition of T into an almost direct product of split and anisotropic subtori.For each root α, let g α be the root subspace of g.We identify elements of g with the induced tangent vectors at (iX, 1) coming from the canonical vector fields.Then under the Hermitian form , , t is perpendicular to each root space; Lie T a is perpendicular to Lie T s ; g α is perpendicular to g β unless β = −α; , is positive definite on Lie T s and negative definite on Lie T a ; , is isotropic on g α ⊕ g −α if α is not imaginary; and , is positive (resp.negative) definite on g α if α is noncompact imaginary (resp.compact imaginary).
A priori, the signature of , is only constant on each connected component of M , but in fact more is true: Corollary 6.5.The signature of , is constant on M .
Proof.Since (i0, 1) ∈ M , there exists a connected neighborhood U of 0 in g R such that iU × G R ⊂ M .It follows that , has constant signature on iU × G R (note that while G R need not be connected, this is irrelevant since , is G R -invariant).We must show that the signature of , on any connected component of M is the same as on iU × G R .Without loss of generality we may choose (iX, r) ∈ M such that X is regular semisimple in g, which is to say that X ∈ t R for a (unique) Cartan subalgebra t R ⊂ g R .We may find s ∈ R * sufficiently small that sX ∈ U , and of course sX ∈ t R is still regular semisimple.However by 6.4, the signature at (iY, r) ∈ M (for Y regular semisimple) depends only on attributes of the unique real Cartan subalgebra containing Y and of its root system.Hence the signature of , at (iX, r) is the same as the signature of , on iU × G R .
Corollary 6.6.The signature of , equals the signature of the Hermitian form Z, W → κ(Z, W ) on g, which equals the signature of the (real) symmetric bilinear form Z, W → κ(Z, W ) on g R .
Proof.By 6.5, it is enough to check the result at a single point of M , and at the point (i0, 1), Z, W = κ(Z, W ). The second statement is a simple linear algebra fact.

Alternative Vector Fields and Comparison with Fels' Work
Let (iX, r) ∈ M .Since G R ×G R acts on M , any pair (Y 1 , Y 2 ) ∈ g R ×g R produces a tangent vector at (iX, r) (indeed it produces an "orbital vector field" on all of ig R × G R ).It is easy to see that (Y 1 , Y 2 ) and (Y 1 , Y 2 ) produce the same tangent vector at (iX, r) if and only if (Y 1 , Y 2 ) = (Y 1 + A, Y 2 + Ad −1 r A) for some A in the centralizer in g R of X.Given any V in this centralizer, we obtain a (nonorbital) tangent vector to M at (iX, r), namely d dt t=0 (iX + itV, r) (however this need not be extendable to a vector field on M ).By dimension count, any tangent vector to M at (iX, r) can be obtained from a combination of orbital and transverse vectors; namely, given (Y 1 , Y 2 , V ) as above, we obtain the tangent vector d dt t=0 Ad e tY 1 (iX + itV ), e tY1 re −tY2 , and every tangent vector can be obtained in this way.The relationship to canonical vectors is: Proposition 7.1.At (iX, r) (with p = e iX ), the tangent vector coming from the triple (Y 1 , Y 2 , V ) coincides with the canonical vector ) and (Y 3 , Y 4 , V 2 ) as above.Let Z, W ∈ g be the corresponding canonical vectors (only valid for the point (iX, r)!).It follows from 2.4(2) and 5.1(1,2,4) that: Using 5.1(1,3,5), we can separate easily the real and imaginary parts of Z, W : We recall some facts about moment maps (see [CG, Chapter 1], [HW]).Suppose that a Lie group K acts symplectically on a symplectic manifold (N, ω).There is a map sending smooth functions on N to symplectic (=locally Hamiltonian) vector fields on N .Since G acts symplectically, there is also a map sending Documenta Mathematica 5 (2000) 595-611 each element of k to a symplectic vector field.The action of K is said to be Hamiltonian if there is a Lie algebra homomorphism H from k to smooth functions on N which makes a commutative triangle with the other two maps.The associated moment map µ : N → k * is the map sending n ∈ N to the linear function on k given by x → H x (n).If the manifold in question is a cotangent bundle, with canonical 1-form λ and ω := dλ and with K acting on the base space, then (N, ω) is Hamiltonian, with H sending x ∈ k to the contraction of λ with the vector field coming from the infinitesimal action of x.
X, e tY1 re −tY2 , so by the remark at the beginning of §4, we have We identify g R with g * R via the Killing form.We have proved: Theorem 7.4.Relative to the symplectic form ω, the moment map µ Another easy consequence of 7.4 is: The formula for ω( Z, W ) in (7.3) is essentially due to Gregor Fels [F].Here we recall his construction in [F] of a pseudokähler form on certain complex manifolds and relate the construction to the one in this paper.(We have changed notation slightly from [F].) Let G R ⊂ G as usual, and let t be a Cartan subalgebra of g that is stable under complex conjugation.Let t R denote the regular semisimple elements of t R .Define N = {(n, n) : n ∈ N G R (T R )} ⊂ G R × G R .Since N acts on t R , we have the usual twisted product (G R × G R ) * N it R .It is easy to see that the map Θ : given by [(r 1 , r 2 ), iX] → Ad r1 (iX), r 1 r −1 2 is well-defined, injective, and G R × G R -equivariant, with everywhere nonsingular differential.Moreover, the map forms one leg of a commutative triangle with the maps It follows that there is a complex structure on (G R × G R ) * N it R that agrees with the ones on G and on M .Given the point v = [(r 1 , r 2 ), iX] ∈ (G R × G R ) * N it R , one can construct (any) tangent vector as d ds s=0 of the curve s → [(r 1 e sY1 , r 2 e sY2 ), iX + siY 3 ], where Y 1 , Y 2 ∈ g R and Y 3 ∈ t R .The J-invariant 2-form on (G R × G R ) * N it R constructed in [F] arises as dθ, where θ is the 1-form which sends the above tangent vector to κ(X, Y 1 −Y 2 ).However, one can show that Θ induces an identification between the 1-forms θ and λ, and hence Θ induces an identification between (the restriction of) the pseudokähler form in the present paper and the one in [F].
denote the Lie algebra of G, with Killing form κ. Given Y ∈ g, the left-and right-invariant vector fields on G generated by Y are denoted g → dl g Y and g → dr g Y .We can identify the cotangent bundle T * (G R ) with ig R × G R ; namely, from X ∈ g R and k ∈ G R , we obtain the 1-form at k that sends dr k Y to κ(X, Y ), where Y ∈ g R .The action of G R × G R on G R (by left/right translation) induces an action on T * (G R ), which in the above identification gives the following action of is a topologically closed, smooth, Int G R -stable submanifold of G of real dimension equal to the complex dimension of G.The subset B(G) := {h • h −1 : h ∈ G} ⊂ Z(G) coincides with the connected component of Z(G) containing 1. Since e iX = e iX/2 • e iX/2 −1 , we have exp(ig R ) ⊂ B(G).§2.The Complex Manifold M and Canonical Vector Fields Proof of 2.3.Tangent vectors at pr which are in the image of the differential of the multiplication map at (p, r) are exactly those of the form d dt t=0 pe tZ e tY r = d dt t=0 pe t(Z+Y ) r, where Z = −Ad p Z and Y = Y .Hence (Z, Y ) is in the kernel of the differential exactly when Z = −Y , which is possible for Z exactly when Z is real, meaning Ad p Z = −Z.