Loops in Noncompact Groups of Hermitian Symmetric Type and Factorization

In studies of Pittmann, we showed that a loop in a simply connected compact Lie group has a unique Birkhoff (or triangular) factorization if and only if the loop has a unique root subgroup factorization (relative to a choice of a reduced sequence of simple reflections in the affine Weyl group). In this paper our main purpose is to investigate Birkhoff and root subgroup factorization for loops in a noncompact semisimple Lie group of Hermitian symmetric type. In literature of caine, we showed that for an element of, i.e. a constant loop, there is a unique Birkhoff factorization if and only if there is a root subgroup factorization. However for loops in, while a root subgroup factorization implies a unique Birkhoff factorization, the converse is false. As in the compact case, root subgroup factorization is intimately related to factorization of Toeplitz determinants.


Introduction
Finite dimensional Riemannian symmetric spaces come in dual pairs, one of compact type and one of noncompact type. Given such a pair, there is a diagram of finite dimensional groups The main purpose of this paper is to investigate Birkhoff (or triangular) factorization and "root subgroup factorization" for the loop group of 0 G  , assuming 0 G  is of Hermitian symmetric type so that X 0 and X are Hermitian symmetric spaces. Birkhoff factorization is investigated in studies of Caine and Wisdom [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], from various points of view. In particular Birkhoff factorization for is developed in Chapter 8 of Wisdom [15], using the Grassmannian model for the homogeneous space / LU U   . Root subgroup factorization for generic loops in U  appeared more recently in literature of Pickrell [11] (for = (2) U SU  , the rank one case) and Pittmann [13]. The Birkhoff decomposition for LG  , is far more complicated than for LU  . With respect to root subgroup factorization, beyond loops in a torus (corresponding to imaginary roots), in the compact context the basic building blocks are exclusively spheres (corresponding to real roots), and in the Hermitian symmetric noncompact context the building blocks are a combination of spheres and disks. This introduces additional analytic complications, and perhaps the main point of this paper is to communicate the problems that arise from noncompactness.
For g LU ∈  , the basic fact is that g has a unique triangular factorization if and only if g has a unique "root subgroup factorization" (relative to the choice of a reduced sequence of simple reflections in the affine Weyl group). This is also true for elements of 0 G  (constant loops); [4]. However, somewhat to our surprise, this is far from true for loops in 0 G  .
Relatively little sophistication is required to state the basic results in the rank one noncompact case. This is essentially because (in addition to loops in a torus) the basic building blocks are exclusively disks, and there is essentially a unique way to choose a reduced sequence of simple reflections in the affine Weyl group, so that the dependence on this choice can be suppressed.

The Rank 1 Case
We consider the data determined by the Riemann sphere and the Poincaré disk. For this pair, the diagram (0. There is a similar set of implications for g 2 ∈ L fin SU (1,1) and the following statements: where c and d are polynomials in z of order n and n-1, respectively, with c(0) = 0 and d(0) > 0.
(II.2) g 2 has a "root subgroup factorization" of the form In general we do not know how to describe the connected component in the first and third conditions. The following example shows how disconnectness arises in the simplest nontrivial case.
Example 0.2. Consider the case n = 2 and g 2 as in II.3 with It is straightforward to check that this g 2 does indeed have values in SU (1,1). In order for 2 2 > 0 a , there are two possibilities: the first is that both the numerator and denominator are positive, in which case there is a root subgroup factorization (with , and the second is that both the top and bottom are negative, in which case root subgroup factorization fails (because when there is a root subgroup factorization, we must have | ζ 1 |,| ζ 2 | < 1).
In order to formulate a general factorization result, we need a C ∞ version of Theorem 0.1.
Theorem 0.2. Suppose that g 1 ∈ LSU (1,1). The following conditions are equivalent: where a and b are holomorphic in ∆ and have C ∞ boundary values, with a(0) > 0.
(I.3) g 1 has triangular factorization of the form where y is holomorphic in ∆ with C ∞ boundary values, a 1 > 0, and the third factor is a matrix valued polynomial in z which is unipotent upper triangular at z = 0.
Similarly if g 2 ∈ LSU(1,1), the following statements are equivalent: where c and d are holomorphic in ∆ and have C ∞ boundary values, with c(0) = 0 and d(0) > 0.
(II.3) g 2 has a triangular factorization of the form The following theorem is the analogue of Theorem 0.2 of [11] (the notation is taken from Section 1 of [11], and reviewed below the statement of the theorem).
Theorem 0.3. Suppose g ∈ LSU(1,1) (0) , the identity component. Then g has a unique "partial root subgroup factorization" of the form  x ζ , and this loop will often not satisfy the condition 2 1 2 | |< 1 x z x z + on S 1 . In this case g will not have a partial root subgroup factorization in the sense of Theorem 0.3.
The group LSL(2,) has a Birkhoff decomposition (2, ) where W (an affine Weyl group, and in this case the infinite dihedral group) is a quotient of a discrete group of unitary loops (the reflections corresponding to the two simple roots for the Kac-Moody extension of sl(2,). The set l has smooth boundary values on S 1 , and u has smooth boundary values on S 1 . If w = 1, the generic case, then we say (as in Section 1 of [11]) that g has a triangular factorization, and in this case the factors are unique.
Next, let LSU(1,1) (n) denote the connected component containing and empty otherwise (refer Section 8.4 of [15]); in particular this intersection is contractible to w, modulo multiplication by T  . Based partly on the finite dimensional results in [4], one might expect the following to be true: (1) Modulo T  , it should be possible to contract should be empty unless w is represented by a loop in SU(1,1). (2) To summarize one surprise, the set of loops having a root subgroup factorization is properly contained in the set of loops in the identity component which have a triangular factorization which, in turn, is a proper subset of the identity component of LSU (1,1). It seems plausible that all of the intersections (1,1) ( ) LSU n w Σ are nonempty, and topologically nontrivial. Unfortunately we lack a geometric explanation for why these intersections are so complicated.

Toeplitz determinants
The group LSU(1,1) acts by bounded multiplication operators on the Hilbert space H := L 2 (S 1 ; 2 ). As in literature of Widom [15], this defines a homomorphism of LSU(1,1) into the restricted general linear group of H defined relative to the Hardy polarization H = H + ⊕ H − , where H + is the subspace of boundary values of functions in H 0 (∆,) and H − is the subspace of boundary values of functions in H 0 (∆ * , ). For a loop g, let A(g) (respectively, A 1 (g)) denote the corresponding Toeplitz operator, i.e., the compression of multiplication by g to H + (resp., the shifted Toeplitz operator, i.e. the compression to ). It is well known that A(g) A(g −1 ) and A 1 (g) A 1 (g −1 ) are determinant class operators (i.e., of the form 1+ trace class).
Theorem 0.4. Suppose that g ∈ LSU(1,1) (0) has a root subgroup factorization as in part (b) of Theorem 0.3. Then and if g = lmau is the triangular factorization as in (0.5) (with w = 1), then ζ ∞ are the zero sequences (the abelian case), the first formula specializes to a result of Szego and Widom (Theorem 7.1 of [16]). Estelle Basor pointed out to us that this result, for g as in (0.3), can be deduced from Theorem 5.1 of [16].

Additional motivation
There is a developing analogue of root subgroup factorization for the group of homeomorphisms of a circle, a group which (in some ways) is similar to a noncompact type Lie group [12]; there are other analogues as well [1]. It is important to identify potential pitfalls. In this paper our primary contribution is perhaps to identify what can go wrong with Birkhoff and root subgroup factorization for loops into a noncompact target; these lessons are potentially valuable in other contexts.
From another point of view, it is expected that root subgroup factorization is relevant to finding Darboux coordinates for homogeneous Poisson structures on LU  and 0 LG  [10]. As of this writing, this is an open question.

Plan of the paper
This paper is essentially a sequel to studies of Pittmann and Pressley [4,13]. We will refer to the latter paper as the 'finite dimensional case', and we note the differences as we go along. Section 1 is on background for finite dimensional groups (which is identical to [4]) and loop groups. In section 2 we consider the intersection of the Birkhoff decomposition for LG  with 0 LG  . Unfortunately for loops in 0 G  , there does not exist an analogue of "block (or coarse) triangular decomposition", a key feature of the finite dimensional case. Consequently there does not exist a reduction to the compact type case, as in finite dimensions. One might still naively expect that there could be a relatively transparent way to parameterize the intersections of the Birkhoff components with 0 LG  (as in the finite dimensional case, and in the case of loops into compact groups, e.g., using root subgroup factorization). But these intersections turn out to be topologically nontrivial. Most of the section is devoted to rank one examples which illustrate this.
In Section 3 we consider root subgroup factorization for generic loops in 0 G  . Our objective in this section is to prove analogues of Theorems 4.1, 4.2, and 5.1 of studies of Pittmann [13], for generic loops in (the Kac-Moody central extension of) 0 LG  (when 0 G  is of Hermitian symmetric type). As in the rank one case above, all of the statements have to be severely modified. The structures of the arguments in this noncompact context are roughly the same as in literature of Pittmann [13], but there are many differences in the details (reflected in the more complicated statements of theorems).

Notation and Background
In this paper, we will make use of the fact that (certain extensions of) loop algebras of complex semisimple Lie algebras and finite dimensional complex semisimple Lie algebras fit into the common framework of Kac-Moody Lie algebras. To distinguish data associated the finite dimensional Lie algebras from the analogous information for the infinite dimensional loop algebra of such a Lie algebra, we will adhere to a convention of Kac and label the data associated with finite dimensional data by an overhead dot.

Finite dimensional groups and algebras
We consider the data (0.1) determined as follows from a compact Hermitian symmetric space X  . We consider the isometry group of X  and let U  denote the universal covering group. Then U  is a simply connected compact group and we let K  be the stability subgroup of a point in X  , so that Remark 1.1. The notation * ( ) − ⋅ for the Cartan involution fixing U  inside of G  is suggestive of the matrix operation of inverse conjugate transpose which fixes SU(n) inside of SL(n,). Likewise, we will use (⋅) * to denote the operation Thus, we obtain the diagram of finite dimensional groups (0.1). Correspondingly, we obtain an analogous diagram of finite dimensional Lie algebras 0  g ,  k ,  g ,  u , and we use Θ  and (⋅) * to also denote the corresponding infinitesimal involutions. Let 0 = + We This determines a choice of positive roots for the action of  h on  g . Let ±  n denote the sum of the positive (resp. negative) root spaces. Then (c) when γ is of noncompact type then γ ι  restricts to We denote the corresponding group homomorphism by the same symbol. Note that if γ is of noncompact type, then γ ι  induces an embedding of the rank one diagram (0.2) into the finite dimensional group diagram (0.1). For each simple positive root γ , we use the group homomorphism to set and obtain a specific representative for the associated simple reflection   corresponding to γ . (We will adhere to the convention of using boldface letters to denote representatives of Weyl group elements).
The affine Weyl group for  g is the semidirect product W T known as the fundamental alcove. Since C 0 will play the role for an infinite dimensional group G extending LG  that C  plays for the finite dimensional group G  , we purposely omit an overhead dot from the label C 0 . g is a subalgebra of L g with respect to the the point-wise bracket. There is a universal central extension

Loop algebras and extensions
as a vector space, and The smooth completion of the untwisted affine Kac-Moody Lie algebra corresponding to  g is where the derivation d acts by L g (and similarly for  as in the compact case [13]).
We identify  g with the constant loops in L g . Because the extension is trivial over  g , there are embeddings of Lie algebras The involution Θ  on  g induces an involution on L g by post-composition. We extend this to an involution Θ on  = L g g by declaring that Θ(c) = c, and similarly extend it to  L g by declaring that Θ(d) = d.
, respectively, are triangular decompositions. The simple positive roots for the pair  , and the α  j are extended to d+h by requiring satisfy the sl(2,) -commutation relations, and e θ  is a highest root vector for  g . The fundamental dominant integral functionals on h are Λ j , j = 0,..,r.

Loop groups and extensions
LG  ).

Proposition 1.3. Π induces a central circle extension
(and similarly for unitary loops as in literature of Pittmann [13]).
Proof. This follows from Proposition 1.2.
Let  = G LG  and let N ± denote the subgroups corresponding to n ± . Since the restriction of Π to N ± is an isomorphism, we will always identify N ± with its image, e.g., l ∈ N + is identified with a smooth loop in G  having a holomorphic extension to ∆ satisfying (0) l N + ∈  . Also, set T = exp(t) and A = exp(a).
As in the finite dimensional case, for  (a) N ± are stable with respect to Θ, whereas N ± are interchanged by (⋅) * . If , and    1 1/2 | |( ):=| ( )|= ( ( ) ( )) . Proof. (a) and (b) follow from the compatibility of the triangular factorization with respect to Θ and u. The first part of (c) follows from the fact that the induced extension  0 LG  is unitary. The formula 1.8 in (c) follows from the fact that if λ∈, then where l is the level.

A note on the rank one case
In this subsection we will freely use the notation in Section 1 of [11] and [15] (as in section 1 of [11], we denote the Toeplitz and shifted Toeplitz operators by A and A 1 , respectively).
In the rank one case σ 0 and σ 1 can be concretely realized as "regularized Toeplitz determinants." In the notation of section 6.6 of [15], a concrete model for the central extension is  In general, we can present a given w ∈W as

Reduced sequences in the affine weyl group
We let Inv(w) denote the inversion set of w, i.e., the set of positive roots which are mapped to negative roots by w. Remark 1.3. In the finite dimensional context [4], the root subgroup factorization of generic elements of 0 G  depended on a reduced expression for 0 w  , the longest element of the Weyl group W  , i.e., a finite reduced word in simple reflections. In this infinite dimensional context, where there is no longest element of W, we must allow the possibility that root subgroup factorization of generic elements will depend on a possibly infinite sequence Remark 1.4. In the rank one case, there are only two possible reduced sequences since W is the infinite dihedral group. As a result, there are only two forms for the root subgroup factorization of generic elements of LSU (1,1). This is the reason for the structure of the theorems stated in the Introduction, involving two sets of analogous implications. In the higher rank setting, however, there are infinitely many forms the factorization could take.  γ ∞ , i.e., γ s+l = γ s for each s. Through the affine action, the sequence of reflections applied to the fundamental alcove C 0 determines a non-terminating walk through the alcoves in  a . In these terms, affine periodicity of the sequence =1 ( ) j j r ∞ means that the walk from step l+1 to 2l is the original walk up to step l translated by , and so on.
We now recall Theorem 3.5 of [13] (this is what we will need in Section 3 for root subgroup factorization of generic loops in 0 G  ).

Contrast with finite dimensions
In literature of Caine [4] we considered 0 G  (constant loops). The key fact (depending on the Hermitian type assumption) was that where each of the two summands is a subalgebra, but the sum is not a Lie algebra (let alone an abelian ideal in a parabolic subalgebra). The fundamental difficulty is that in the finite dimensional case N +  is a nilpotent group, and hence whenever the Lie algebra is a sum of subalgebras, there is a corresponding global decomposition at the group level. However, in the loop case N + is a profinite nilpotent group, and the corresponding result is not true, e.g., a holomorphic map from from the disk to the Lie algebra has a pointwise triangular decomposition, but pointwise triangular factorization fails very badly at the group level. For example, the SL(2, ) -valued holomorphic does not have a pointwise triangular factorization because the (1,1) entry vanishes at z = 1/2.

Compact vs noncompact type roots in g
As in the finite dimensional setting, a root of h on g is said to be of compact type if the corresponding root space belongs to k, and said to be of noncompact type if the corresponding root space belongs to p  . Here Remark 1.6. In rank one, the compact type roots are the imaginary roots and the noncompact type roots are the real roots. This is yet another special feature of the rank one case.

The basic framework and notation
In the remainder of the paper we will mainly be concerned with the loop analogue of (0. LG  , and := K LK   , the central circle extension of LK  . There is a corresponding diagram of Lie algebras, where the Lie algebra of G is = L   g g, and so on.
It will often happen that we can more simply work at the level of loops, rather than at the level of central extensions. We will often state results, for example, in terms of G, but in proving results it is often possible and easier to work with LG  .

Birkhoff Decomposition for Loops
By definition the Birkhoff decomposition of If we fix a representative w ∈ N U (T) for w ∈ W, then each  As we stated in the introduction (where we focused on the rank one case), our original expectation was that each of these components would be (modulo a torus) contractible to w. Our main objective in this subsection is to provide examples in the rank one case, for the identity component, which illustrate why this is not true.
Proof. For any g ∈ LSU(1,1) there is a pointwise polar decomposition , and λ : S 1 →S 1 . We can always multiply g on the left (right) by something in B − (B + , respectively) without affecting the question of whether g has a triangular factorization. For example in determining whether g has a triangular factorization, we can ignore the factor exp(ψ − + ψ 0 ) in λ, because this can be factored out on the left. We will use this observation repeatedly (note that we can recover ψ − from ψ + , and the zero mode is inconsequential).
There is a factorization of To obtain g we have to multiply this on the left by λ. It follows after some calculation that g will have a triangular factorization if and only if has a triangular factorization.
At this point, to simplify notation, we let has a triangular factorization. Note that the (2,2) entry of the right hand side equals It is easy to find ψ + and b 2 such that there does exist a nonzero F satisfying this condition.
Factor l as Then g will have the form where L′∈N − . Consequently to find the Birkhoff factorization for g, it suffices to find the factorization for the triangular matrix valued function Remark 2.1. What we are doing here is factoring N − as N − ∩ wN − w −1 times N ∩ wN + w −1 . So this is very general. The problem of understanding Birkhoff factorization for triangular matrix valued functions is considered in literature of Clancey [5].

Example 2.2
When n = 1, we could take g 0 = g 1 in Theorem 0.1. Then is invertible.
Proof. The Fredholm indices for both operators are zero, so we need to check the kernels.

Part (a): Suppose that
This equation implies h k = 0 for k ≥ n. These equations have the matrix form is the vector of coefficients of f (resp. h) and A′ is the n × n Toeplitz matrix in (2.5). This implies part (a).
The proof of part (b) is similar.
When c 0 , c 1 ≠ 0, there is a triangular factorization (because A and A 1 are invertible), In this case When c 1 → 0 this "degenerates" to a Birkhoff factorization In this case When c 0 → 0 this "degenerates" to a Birkhoff factorization 2 1 1 In this case

G
Our objective in this section is to prove analogues of Theorems 4.1, 4.2, and 5.1 of [13], for generic loops in 0 G  (which is always assumed to be of Hermitian symmetric type). The structure of the proofs in this noncompact context is basically the same as in literature of Pittmann [13]. But there are important differences. In order to obtain formulas for determinants of Toeplitz operators, as in Theorem 0.4, we have to work with the central extension  LG  .
Throughout this section we choose a reduced sequence =1 { } j j r ∞ as in Theorem 1.1, part (a). We set w j = r j …r 1 and and for n > 0 As in studies of Caine [4], for ζ ∈  , let   where g(η j ) = k(η j ) for some η j ∈ (resp. g(η j ) = q(η j ) for some η j ∈∆) when τ j is a compact type (resp. non-compact type) root.   In the course of the following proof of Theorem 3.1, we will prove a version of this conjecture, in the rank one case, which completes the proof of Theorem 0.1 (Remark 3.2 below).
Proof. The two sets of implications are proven in the same way. We consider the second set.
We first want to argue that (II.2) implies (II.3). We recall that the subalgebra is spanned by the root spaces corresponding to negative roots −τ j , j=1,..,n. The calculation is the same as in the proof of Theorem 2.5 in [4]. In the process we will also prove the product formula for a 2 .
The equation (3.1) implies that is a triangular factorization. Here, ( ) = ( ) j j ζ ζ ± a a and the plus/minus case is used when τ j is a compact/noncompact type root, respectively. The key point is that Insert this calculation into (3.4). We then see that g (2) has a triangular factorization g (2) = l (2) a (2) u (2) , where (the last equality holds because a two dimensional nilpotent algebra is necessarily commutative).
To apply induction, we assume that g (n-1) has a triangular factorization g (n-1) = l (n-1) a (n-1) u (n-1) with    Remark 3.2. In reference to Conjecture 6.1, we observe that the preceding calculation shows that we have a map (using the notation we where each ζ j ranges over either the complex plane or a disk, depending on whether the jth root is of compact or noncompact type. The calculation above also shows that the map is 1-1 and open. We claim that the image of this map is closed in This follows from the product formula for a 2 , which shows that as the parameters tend to the boundary, the triangular factorization fails. This implies that the image of the map is the connected component which contains l 2 = 1. This proves the implication (II.2) implies (II. 3) in the special case of Theorem 3.1, because n is fixed in the statement of that theorem, but this does not complete the proof of Conjecture 6.1. The difficulty is that we do not know how to formulate statements (I.1) and (II.1) in the general case in a way that regards n as fixed.
It is obvious that (II.3) implies (II.1). In fact (II.3) implies a stronger condition. If (II.3) holds, then given a highest weight vector v as in (II.1), corresponding to highest weight Λ  , then is holomorphic in ∆ and nonvanishing at all points. However we do not need to include this nonvanishing condition in (II.1), in this finite case.
It remains to prove that (II.1) implies (II.3). Because  2 g is determined by g 2 , as in Lemma 1.4, it suffices to show that g 2 has a triangular factorization (with trivial T  component). Hence we will slightly abuse notation and work at the level of loops in the remainder of this proof.
To motivate the argument, suppose that g 2 has triangular factorization as in (II.3). Because 2 (0) u N + ∈  , there exists a pointwise G  -triangular factorization ( ) = ( ( ) ) ( ( ) ) ( ( ) ) ( ) . g z l u z d u z a a u u z a l z This is a pointwise G  -triangular factorization of 1 2 g − , which is certainly valid in a punctured neighborhood of z = 0. The important facts are that (1) the first factor in (3.10) = g g − in a highest weight representation. Then (3.14), together with (II.1), implies the claim.
The factorization (3.12) is unobstructed. Thus it exists. We can now read the calculation backwards, as in (3.13), and obtain a triangular factorization for g 2 as in (II.3) (initially for the restriction to a small circle about 0; but because g 2 is of finite type, this is valid also for the standard circle). This completes the proof.
In the C ∞ analogue of Theorem 3.1, it is necessary to add further hypotheses in parts I. 1 Consider the following three statements: , and for each complex irreducible representation V(π) for G  , with lowest weight vector φ ∈ V(π), π(g 1 ) −1 (φ) has holomorphic extension to ∆, is nonzero at all z ∈ ∆, and is a positive multiple of v at z = 0.  where g(η j ) = k(η j ) for some η j ∈ (resp. g(η j ) = q(η j ) for some η j ∈∆) when τ j is a compact type (resp. non-compact type) root and the sequence where g(ζ j ) = k(ζ j ) for some ζ j ∈  (resp. g(ζ j ) = q(ζ j ) for some ζ j ∈ ∆) when τ j is a compact type (resp. non-compact type) root and the sequence =1 ( ) j j ζ ∞ is rapidly decreasing.
3.  2 g has triangular factorization of the form  In Remark 3.3, at the end of the following proof, we will indicate how we envision proving this conjecture. The issue in this C ∞ context involves analysis, and we are not as confident in the truth of this Conjecture 3.2.
Proof. The two sets of equivalences and implications are proven in the same way. We consider the second set.
Suppose that (II.1) holds. To show that (II.3) holds, it suffices to prove that g 2 has a triangular factorization with l 2 of the prescribed form (Lemma 1.4). By working in a fixed faithful highest weight representation for  g , without loss of generality, we can suppose 0 G  is a matrix subgroup of SL(n,) (where +  n consists of upper triangular matrices). We will assume that this representation is the complexified adjoint representation, or some subrepresentation of the exterior algebra of the adjoint representation, so that we can suppose that 0 G  fixes a (indefinite) Hermitian form (in the case of the adjoint representation, this is derived from the Killing form).
For the purposes of this proof, we will use the terminology in Section 1 of literature of Pickrell [11]. We view 2 0 g LG ∈  as a multiplication operator on the Hilbert space  , the subspace of functions in  with holomorphic extension to ∆. To show that g 2 has a Birkhoff factorization, we must show that A(g 2 ) is invertible (Theorem 1.1 of [11]).
Let C 1 ,..,C n denote the columns of 1 2 g − , and let * * 1 ,.., n C C denote the rows of g 2 . We can regard these as dual bases with respect to the pairing given by matrix multiplication, i.e., * = i j ij C C δ .
The hypothesis of (II.1) implies that both C 1 and * n C have holomorphic extensions to ∆ (in the latter case, by considering the dual representation). Now suppose that f + ∈  is in the kernel of A(g 2 ). Then  (3.17). This implies that for z ∈ S 1 , f (z) is a linear combination of the n − 1 columns C j (z), j<n. We write The vectors C 1 ∧ .. ∧ C j extend holomorphically to ∆, and never vanish, for any j, by (II.1) (by considering the representation ∧ j ( n )). Since f also extends holomorphically, this implies that n−1 has holomorphic extension to . Now Since the right hand side is holomorphic in ∆, by (3.17) (for j = n−1) λ n−1 vanishes identically. This implies that in fact f is a (pointwise) linear combination of the first n−2 columns of 1 2 g − . Continuing the argument in the obvious way (by next wedging f with C 1 ∧ .. ∧ C n−3 to conclude that λ n−2 must vanish), we conclude that f is zero. This implies that ker(A(g 2 )) = 0. Since G  is simply connected, (A(g 2 ) has index zero. Hence (A(g 2 ) is invertible. This implies (II.3).