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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Loops in Noncompact Groups of Hermitian Symmetric Type and Factorization

Caine A* and Pickrell D

California State Polytechnic University Pomona, Mathematics and Statistics, 3801 W, Temple Ave. Pomona, CA 91768, USA

Corresponding Author:
Caine A
Professor, California State Polytechnic University Pomona
Mathematics and Statistics, 3801 W
Temple Ave. Pomona, CA 91768, USA
Tel: +19098697659
E-mail: [email protected]

Received date: July 29, 2015 Accepted date: October 30, 2015 Published date: November 05, 2015

Citation: Caine A, Pickrell D (2015) Loops in Noncompact Groups of Hermitian Symmetric Type and Factorization. J Generalized Lie Theory Appl 9:233. doi: 10.4172/1736-4337.1000233

Copyright: © 2015 Caine A, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In previous work with Pittmann-Polletta, 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 type semisimple Lie group of Hermitian symmetric type. In previous work we showed that for a constant loop there is a unique Birkhoff factorization if and only if there is a root subgroup factorization. However for loops, 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.

Keywords

Noncompact groups; Birkhoff factorization; Weyl group

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

image (0.1)

where image is the universal covering of the identity component of the isometry group of the compact type symmetric space imageis the complexification of image, and imageis a covering of the isometry group for the dual noncompact symmetric spaceimage.

The main purpose of this paper is to investigate Birkhoff (or triangular) factorization and “root subgroup factorization" for the loop group of imageassuming image is of Hermitian symmetric type so that X0 and X are Hermitian symmetric spaces. Birkhoff factorization is investigated in studies of Caine and Wisdom [1-15], from various points of view. In particular Birkhoff factorization for image is developed in Chapter 8 of Wisdom [15], using the Grassmannian model for the homogeneous space image.Root subgroup factorization for generic loops in imageappeared more recently in literature of Pickrell [11] (for image = SU(2) , the rank one case) and Pittmann [13]. The Birkhoff decomposition for image , i.e., the intersection of the Birkhoff decomposition for imageis far more complicated than for image.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 image,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 image(constant loops); [4]. However, somewhat to our surprise, this is far from true for loops in image.

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 (0.1) becomes

image (0.2)

Let LfinSL(2,C) denote the group consisting of maps S1 → SL(2,C) having finite Fourier series, with pointwise multiplication. The subset of those functions having values in SU(1,1) is then a subgroup, denoted LfinSU(1,1).

Example 0.1 For each image the function image defined by is in LfinSU(1,1)

image (0.3)

LfinSU(2) and LfinSU(1,1) are dense in the smooth loop groups LSU(2) := C(S1,SU(2)) and LSU(1,1) := C(S1,SU(1,1)), respectively. This is proven in the compact case in Proposition 3.5.3 of [15], and the argument applies also for SU(1,1), taking into account the obvious modifications.

For a Laurent series imageIf Ω is a domain on the Riemann sphere, we write H0(Ω) for the vector space of holomorphic scalar valued functions on Ω. If ∈ H0(Δ), then f *∈ H0(Δ*), where Δ* denotes the open unit disk at ∞.

Theorem 0.1 Suppose that g1 ∈ LfinSU(1,1) and fix n > 0. Consider the following three statements:

(I.1) g1 is of the form

image

where a and b are polynomials in z of order n-1 and n, respectively, with a(0) > 0.

(I.2) g1 has a “root subgroup factorization” of the form

image

for some sequence image is the function in Example 0.1.

(I.3) g1 has triangular factorization of the form

image

where a1 > 0, the third factor is a matrix valued polynomial in z which is unipotent upper triangular at z = 0.

Statements (I.1) and (I.3) are equivalent. (I.2) implies (I.1) and (I.3). If g1 is in the identity connected component of the sets in (I.1) and (I.3), then the converse holds, i.e., g1 has a root subgroup factorization as in (I.2).

There is a similar set of implications for g2 ∈ LfinSU(1,1) and the following statements:

(II.1) g2 is of the form

image

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) g2 has a “root subgroup factorization” of the form

image

for some sequence imageis the function in Example 0.1.

(II.3) g2 has a triangular factorization of the form

image

where a2 > 0, and the third factor is a matrix valued polynomial in z which is unipotent upper triangular at z = 0.

When g1 and g2 have root subgroup factorizations, the scalar entries determining the diagonal factor have the product form

image(0.4)

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 g2 as in II.3 with image

image

image

and

image

It is straightforward to check that this g2 does indeed have values in SU(1,1). In order for image, 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 image), 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 image).

In order to formulate a general factorization result, we need a C version of Theorem 0.1.

Theorem 0.2. Suppose that image. The following conditions are equivalent:

(I.1) g1 is of the form

image

where a and b are holomorphic in Δ and have imageboundary values, with a(0) > 0.

(I.3) g1has triangular factorization of the form

image

where y is holomorphic in Δ with image boundary values, a1 > 0, and the third factor is a matrix valued polynomial in z which is unipotent upper triangular at z = 0.

Similarly if the imagefollowing statements are equivalent:

(II.1) g2 is of the form

image

where c and d are holomorphic in Δ and have image boundary values, with c(0) = 0 and d(0) > 0.

(II.3) g2 has a triangular factorization of the form

image

where a2>0, x is holomorphic in Δ and has image boundary values, x(0) = 0, and the third factor is a matrix valued function which is holomorphic in Δ, has image boundary values, and is unipotent upper triangular at z = 0.

Let image denote the anti-holomorphic involution of image which fixes SU(1,1); explicitly

image

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 image, the identity component. Then g has a unique “partial root subgroup factorization" of the form

image

where image and g1 and g2 are as in Theorem 0.2, if and only if g has a triangular factorization g = lmau (0.5) below) such that the boundary values of l21 / l11 and u21 / u22 are <1 in magnitude on S1.

The following example shows that the unaesthetic condition on the boundary values is essential.

Example 0.3. Consider g2 as in Theorem 0.1. The loop g= g2* (the Hermitian conjugate of g2around the circle) has triangular factorization

image

If n = 2, then image and this loop will often not satisfy the condition image on S1. In this case g will not have a partial root subgroup factorization in the sense of Theorem 0.3.

The group image has a Birkhoff decomposition

image

where W (an affine Weyl group, and in this case the infinite dihedral group) is a quotient of a discrete group of unitary loops

image

image

where

image

(the reflections corresponding to the two simple roots for the Kac- Moody extension of imageThe set image consists of loops which have a (Birkhoff) factorization of the form

g = l . w .m . a . u, (0.5)

where image

image

l has smooth boundary values on image

and u has smooth boundary values on S1. 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

image

image

and

image

Since SU(1,1) is homotopy equivalent to the torus image,the connected components of LSU(1,1) are homotopy equivalent to the connected components of image which are indexed by winding number. Write imagefor the connected component indexed by an integer n. Then it is known that the intersection imagewhen imageand empty otherwise (refer Section 8.4 of [15]); in particular this intersection is contractible to w, modulo multiplication by image . Based partly on the finite dimensional results in [4], one might expect the following to be true:

(1) Modulo image, it should be possible to contract imagedown to w; in particular imageshould be empty unless w is represented by a loop in SU(1,1).

(2)image

(3) Each imageshould admit a relatively explicit parameterization.

Statements (1) and (2) are definitely false; statement (3) is very elusive, if not doubtful.

Proposition 0.1.

(a) image can be nonempty even if w is not represented by a loop in SU(1,1). For example,imageis nonempty.

(b) imageis properly contained in LSU(1,1)(0).

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 imageare 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 image. 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 image where H+ is the subspace of boundary values of functions in image and H is the subspace of boundary values of functions in image. For a loop g, let A(g) (respectively, A1(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

image).It is well known that A(g) A(g−1) and A1(g) A1(g−1) are determinant class operators (i.e., of the form 1+ trace class).

Theorem 0.4. Suppose that image has a root subgroup factorization as in part (b) of Theorem 0.3. Then

image

imageand if g = lmau is the triangular factorization as in (0.5) (with w = 1), then

image

When imageare 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 imageand image[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 imagewith image. Unfortunately for loops in image, 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 image (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 imageOur 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) image(when image 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 image.We consider the isometry group of image and let imagedenote the universal covering group. Then imageis a simply connected compact group and we let image be the stability subgroup of a point in image , so that image. This determines an involution Θimage which we extend holomorphically to the complexification image . The composition image with the Cartan involution imagefixing image inside of image is then an antiholomorphic involution of image fixing a real formimagewhich is image-stable. The fixed point set of image in image is image and the coset space image is a model for the non-compact Hermitian symmetric spaceimage dual to image .

Remark 1.1. The notation imagefor the Cartan involution fixing imageinside of imageis suggestive of the matrix operation of inverse conjugate transpose which fixes SU(n) inside of imageLikewise, we will use image to denote the operation image

Thus, we obtain the diagram of finite dimensional groups (0.1). Correspondingly, we obtain an analogous diagram of finite dimensional Lie algebras imageand we useimageand image to also denote the corresponding infinitesimal involutions. Let image denote the eigenspace decomposition of imageunder image . Then image is the eigenspace decomposition of image under image .

Choose a Cartan subalgebra image.Then t is a Cartan subalgebra of imagesince imageis of Hermtian symmetric type. The centralizer h of t in g is then a image -stable Cartan subalgebra of the complex semisimple Lie algebra g . Furthermore, image, where image, is the eigenspace decomposition of image with respect to the involution image.

We will write imagefor the Weyl group of the pair imageChoose a Weyl chamber image.This determines a choice of positive roots for the action of image.Let imagedenote the sum of the positive (resp. negative) root spaces. Then

image

is aimage-stable triangular decomposition of g. An important consequence of image-stability is thatimageare interchanged by the action of image.

Let imagedenote the highest root and normalize the Killing form so that (for the dual form) image.For each root α let timageenote the associated coroot. The Hermitian symmetric type assumption, together with the Θ -stability of h , implies that each root space image contained in either image or in image and thus the roots can be sorted into two types. A root imageis of compact type if the root spaceimage is a subset of image and of noncompact type otherwise, i.e., when imageimage . The following proposition is an elementary fact.

Proposition 1.1. For each simple positive root imagethere exists a Lie algebra homomorphism : image which carries the standard triangular decomposition of image into the triangular decomposition imageand:

(a) in any case imagerestricts to a homomorphism image;

(b) when imageis of compact type then imagerestricts to image

(c) when image is of noncompact type then image restricts to image

We denote the corresponding group homomorphism by the same symbol. Note that if image is of noncompact type, then imageinduces an embedding of the rank one diagram (0.2) into the finite dimensional group diagram (0.1). For each simple positive root image, we use the group homomorphism to set

image (1.1)

and obtain a specific representative for the associated simple reflection image corresponding to image.(We will adhere to the convention of using boldface letters to denote representatives of Weyl group elements).

Remark 1.2. Throughout this paper we regard the homomorphism image corresponding to a simple positive root γ as fixed. If η is another positive root, then there is a Weyl group element w such that image by choosing a representative imagefor w, we obtain a homomorphism imagewith the same properties as in the proposition. This homomorphism will depend on the choice of w and its representative w, but the dependence will be relatively insignificant in this paper.

Let imagedenote the simple positive roots and write imagefor the corresponding coroots. Thenimage g form a basis for image and the dual basis elements image are the fundamental weights. For the coroot lattice, we write

image

The affine Weyl group for image is the semidirect product image.For the action of image , a fundamental domain is the Weyl chamber image.For the natural affine action ofimage on image , a fundamental domain is the convex set

image

known as the fundamental alcove. Since C0 will play the role for an infinite dimensional group G extending image that imageplays for the finite dimensional group image, we purposely omit an overhead dot from the label C0.

Loop algebras and extensions

Let imageand let imagedenote the subspace of functions with finite Fourier series. Then imageis a subalgebra ofimagewith respect to the the point-wise bracket. There is a universal central extension

image (1.2)

where imageas a vector space, and

image(1.3)

The smooth completion of the untwisted affine Kac-Moody Lie algebra corresponding toimage is

image

where the derivation d acts by image, for image , and [d,c] = 0.

Proposition 1.2. For both imageand image the cocycle imageis real-valued. In particular the affine extension (1.2) induces a unitary central extension

image

and a real formimage(and similarly for as in the compact case [13]).image

We identifyimagewith the constant loops inimage. Because the extension is trivial overimage, there are embeddings of Lie algebras image. The involution imageon imageinduces an involution on image by post-composition. We extend this to an involution imageon image by declaring that image , and similarly extend it to image by declaring that image

Let image, and image. We set imageand image. Then, the decompositions

image (1.4)

where image and image denotes the smooth completion of image, respectively, are triangular decompositions. The simple positive roots for the pair imageare imagewhere

image

imageand the imageare extended to imageby requiring image. The simple coroots are imagewhere

image

For i > 0, the root homomorphism imageis simplyimage followed by the inclusion image. For i = 0

image(1.5)

where imagesatisfy the imagecommutation relations, and imageis a highest root vector for image. The fundamental dominant integral functionals on h are image

Loop groups and extensions

Let imagedenote the universal central imageextension of the smooth loop group image(resp.image)

Proposition 1.3. Π induces a central circle extension

image

(and similarly for unitary loops as in literature of Pittmann [13]).

Proof. This follows from Proposition 1.2

Let G = image and let image denote the subgroups corresponding to image. Since the restriction of Π to imageis an isomorphism, we will always identify imagewith its image, e.g., imageis identified with a smooth loop in imagehaving a holomorphic extension to Δ satisfying image. Also, set T = exp(t) and A = exp(a).

As in the finite dimensional case, for image, there is a unique triangular decomposition

image (1.6)

 

and image is the fundamental matrix coefficient for the highest weight vector corresponding to image.If image

, then because imageprojects to image where

image(1.7)

and the imageare positive integers such that image(these numbers are also compiled in Section 1.1 of [7]).

Proposition 1.4.

(a) imageare stable with respect to Θ, whereas image are interchanged byimage.If imagehas triangular factorizationimageas in (1.6), then

image

and

image

are triangular factorizations.

(b) If image,then imageand image

(c) If image, then image depends only on image, and

image(1.8)

(d) For image and image,g has a triangular factorization if and only if g has a triangular factorization. The restriction of the projection image to elements with image

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 image is unitary. The formula 1.8 in (c) follows from the fact that if image, then

image

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 A1, respectively).

In the rank one case σ0 and σ1 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

image

(here image and H+ is the subspace of boundary values of holomorphic functions on the disk). In this realization

image

Proposition 1.5 For image, using the notation in Proposition 1.4,

image

Proof. This follows from (c) of Proposition 1.4.

Reduced sequences in the affine weyl group

The Weyl group W for image acts by isometries of imageThe action of W on imageis trivial. The affine plane image is W -stable, and this action identifies W with the affine Weyl group imageof image and its affine action on image (Chapter 5 of [15]). In this realization, the simple reflection image is a reflection in image followed by a translation in image , specifically

image (1.9)

In general, we can present a given image as

image

We image 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 subgr oup factorization of generic elements of image depended on a reduced expression for image, the longest element of the Weyl group image, 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 image of simple reflections.

Definition 1. We will say that an infinite sequence image of simple reflections in W is reduced if each partial product imageis a reduced expression for each j.

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.

Lemma 1.1. Let image be a reduced sequence of simple reflections in W and let image denote the sequence of corresponding simple positive roots of g. Then for each n:

(a) the inversion set of the partial product wn is

image

(b) image

Definition 2. A reduced sequence of simple reflections image is affine periodic if, in terms of the identification of W with the affine Weyl group,

1. There exists l such that the partial product image is in image, i.e., acts as a translation on image, and

2. image for all s. For the minimal such l, we will refer to image as the period, and l as the length of the period.

Remark 1.5. The second condition is equivalent to periodicity of the associated sequence of simple roots image i.e.,γs+l =γs for each s. Through the affine action, the sequence of reflections applied to the fundamental alcove image determines a non-terminating walk through the alcoves in a . In these terms, affine periodicity of the sequence image means that the walk from step l+1 to 2l is the original walk up to step l translated byimage , 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 image).

Theorem 1.1.

(a) There exists an affine periodic reduced sequence image of simple reflections such that, in the notation of Lemma 1.1,

image (1.10)

i.e., such that the span of the corresponding root spaces is image. . The period can be chosen to be any point in image.

(b) Given a reduced sequence as in (a), and a reduced expression for image(where image is the longest element of image ), the sequence

image

is another reduced sequence. The corresponding set of positive roots mapped to negative roots is

image

i.e., the span of the corresponding root spaces is image.

Many examples illustrating this theorem appear in the dissertation of Pittmann-Polletta ([?]).

4.6 Contrast with finite dimensions

In literature of Caine [4] we considered image(constant loops). The key fact (depending on the Hermitian type assumption) was that

image

where the latter summand, image, is an abelian ideal in the parabolic subalgebra imageof image. This led to a block (coarse) triangular factorization, which largely reduces the (finite dimensional) Hermitian noncompact case to the compact case.

In the present context there is an analogous decomposition

image

where image is the eigenspace decomposition of g0 under image . In this case

image

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 image 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 imagevalued holomorphic function image 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 imageand said to be of noncompact type if the corresponding root space belongs to image. Here imageand image(so this terminology is perhaps less than ideal).

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 (0.1):

image(1.11)

where image, the (simply connected) central circle extension of image , the (simply connected) central imageextension of image , the central circle extension of image , and image , the central circle extension of image . There is a corresponding diagram of Lie algebras, where the Lie algebra of G is =image , 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 image

Birkhoff Decomposition for Loops

By definition the Birkhoff decomposition of imageis

image (2.1)

If we fix a representative image, then each image has a unique Birkhoff factorization

image (2.2)

As in the finite dimensional case, for fixed image is a stratum (diffeomorphic to the product of the Birkhoff stratum for the flag space G / B+ corresponding to w with N+); refer Theorem 8.7.2 of [15]. We will refer toimage as the “(isotypic) component of the Birkhoff decomposition of G corresponding to w ∈ W.”

One virtue of root subgroup factorization is that it generates many explicit examples of Birkhoff factorizations.

Birkhoff decomposition for

image

 

Given image, define

image

Theorem 2.1 Fix a representative image for w. For image the unique factorization (2.2) induces a bijective correspondence

image

We refer to imageas the isotypic component of the Birkhoff decomposition for U; each component consists of a union of strata permuted by the action of T. The theorem provides an explicit parameterization for these strata. We have recalled this result simply for the sake of comparison. Our primary objective is to investigate the Birkhoff decomposition for image.

Birkhoff decomposition forimage , the identity component

Given image, define

image

and so on.

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.

Proposition 2.1.imageis properly contained in image

Proof. For any imagethere is a pointwise polar decomposition

image

where image

If image, then λ has degree zero, and thus λ has a triangular factorization

image

where imageBecause a is a positive periodic function, it will have a triangular factorization

image

where image.

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 expimage, because this can be factored out on the left. We will use this observation repeatedly (note that we can recover image, and the zero mode is inconsequential).

There is a factorization of

image

as the product

image

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

image

has a triangular factorization.

At this point, to simplify notation, we let image. Note that image. Thus g has a triangular factorization if and only if the loop

image

has a triangular factorization. Note that the (2,2) entry of the right hand side equals image.

We directly calculate the kernel of the Toeplitz operator associated to this loop. We obtain the equations (forimage)

image

We can solve the first equation for f1. The second equation becomes

image

If we set image, then this can be rewritten as

image

If we set image, then we see that there exists a nontrivial kernel if and only if there exists nonzero imagesuch that

image

If we set imageand b2 such that there does exist a nonzero F satisfying this condition.

Example 2.1. image. In other words if

image

where

image

then g is a loop in the identity component of LSU(1,1) and does not have a Riemann-Hilbert factorizaton, hence also does not have a triangular factorization.

Birkhoff decomposition for nonidentity components of image

Consider the rank one case and the problem of finding the Birkhoff factorization for g which is of the form image where g0 is in the identity component and has a known triangular factorization (as for example in Theorem 0.1), and n > 0. Write

image

Factor l as

image

Then g will have the form

image

where imageConsequently to find the Birkhoff factorization for g, it suffices to find the factorization for the triangular matrix valued function

image (2.4)

Remark 2.1. What we are doing here is factoring image. 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 g0 = g1 in Theorem 0.1. Then

image

where image

Lemma 2.1 Fix n>0. For a triangular matrix valued function as in (2.4),

(a) the Toeplitz operator A is invertible if and only if the Toeplitz matrix

image (2.5)

is invertible, and

(b) the shifted Toeplitz operator A1 is invertible if and only if the Toeplitz matrix

image (2.6)

Proof. The Fredholm indices for both operators are zero, so we need to check the kernels.

Part (a): Suppose that

image

is in the kernel of A. Then image, implying image, and

image

This equation implies hk = 0 for k ≥ n. These equations have the matrix form

image

where image(resp.image) 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.

Example 2.3. Suppose n = 1. When c0 ≠ 0 there is a Riemann- Hilbert factorization (because A is invertible)

image

When c0, c1 ≠ 0, there is a triangular factorization (because A and A1 are invertible),

image

In this case image

When c1 → 0 this “degenerates" to a Birkhoff factorization

image

In this caseimage

When c0 → 0 this “degenerates" to a Birkhoff factorization

image

In this case image

When both c0, c1→ 0 this goes to image. In this case image,where in the Weyl group image.

These calculations show that we are obtaining loops in the corresponding strata, despite the fact that neither r0 nor r1 are represented by loops in image. Moreover the conditions on c0, c1 above show that the intersection of the image component with the n = −1 connected component is topologically nontrivial. However we do not know how to quantify this.

Root Subgroup Factorization for Generic Loops in image

Our objective in this section is to prove analogues of Theorems 4.1, 4.2, and 5.1 of [13], for generic loops in image(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 imageas in Theorem 1.1, part (a). We set image and

image

image

and for n > 0

image

As in studies of Caine [4], for image, let image and

image

For image , let image and

image

Generalizations of Theorem 3.1

Theorem 3.1. Suppose that image . Consider the following three statements:

(I.1) image and for each complex irreducible representation V() for image , with lowest weight vector imageimageis a polynomial in z (with values in V), and is a positive multiple of image.

(I.2) image has a factorization of the form

image

where g(ηj) = k(ηj) for some imagewhen τj is a compact type (resp. non-compact type) root.

(I.3) image has triangular factorization of the form image

Then statements (I.1) and (I.3) are equivalent. (I.2) implies (I.1) and (I.3).

Moreover, in the notation of (I.2),

image

Similarly, suppose thatimageConsider the following three statements:

(II.1) image, and for each complex irreducible representation with highest weight vector imageimageis a polynomial in z (with values in V), and is a positive multiple of v at z = 0.

(II.2) image has a factorization of the form

image

for some image

(II.3 ) image has triangular factorization of the form image, where image.

Then statements (II.1) and (II.3) are equivalent. (II.2) implies (II.1) and (II.3).

Also, in the notation of (II.2),

image(3.3)

 

Remark 3.1. Note that we are not making any attempt to characterize the set of l1 that arise in (I.3) (and similarly for the set of l2 in (II.3)).

Conjecture 3.1. If g1 is in the identity connected component of the sets in (I.1) and (I.3), then the converse holds, i.e., 1 g has a root subgroup factorization as in (I.2). If g2 is in the identity connected component of the sets in (II.1) and (II.3), then the converse holds, i.e., g2 has a root subgroup factorization as in (II.2).

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 image is spanned by the root spaces corresponding to negative roots image. 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 a2.

The equation (3.1) implies that

image

image

is a triangular factorization. Here, image and the plus/minus case is used when τj is a compact/noncompact type root, respectively.

Let image. First suppose that n = 2. Then

image

The key point is that

image

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

image

and

image

(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

image (3.6)

for some

image, and

image

We have established this for n1 = 1, 2. For n ≥ 2

image

where image. Now write image

relative to the decomposition

image

Let

image

Then g(n) has triangular decomposition

image

This implies the induction step.

This calculation shows that (II.2) implies (II.3). It also implies the product formula for (3.3) a2.

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 have established above)

image

where each image 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

image

This follows from the product formula for a2, 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 l2 = 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

image (3.8)

implying that image is holomorphic in Δ 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 image is determined by g2, as in Lemma 1.4, it suffices to show that g2 has a triangular factorization (with trivial image 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 g2 has triangular factorization as in (II.3). Because image , there exists a pointwise image -triangular factorization

image (3.9)

which is certainly valid in a neighborhood of z = 0; more precisely, (3.9) exists at a point imageif and only if

image

When (3.9) exists (and using the fact that g2 is defined on imagein this algebraic context),

image

This implies

image(3.10)

This is a pointwise image -triangular factorization of image which is certainly valid in a punctured neighborhood of z = 0. The important facts are that (1) the first factor in (3.10)

image (3.11)

does not have a pole at z = 0; (2) for the third (upper triangular) factor in (3.10), the factorization

image (3.12)

is a image -triangular factorization of image, where we view imageas a loop by restricting to a small circle surrounding z = 0; and (3) because there is an a priori formula for a2 in terms of g2 (refer 1.7), we can recover l2and (the pointwise triangular factorization for) image from (3.10)-(3.12): image (by (3.12)), and

image (3.13)

image

We remark that this uses the fact that g2 is defined in image in an essential way.

Now suppose that (II.1) holds. In particular (II.1) implies that imagehas a removable singularity at z = 0 and is positive at z = 0, for i=1,..,r. Thus image has a pointwise image -triangular factorization as in (3.10), for all z in some punctured neighborhood of z = 0.

We claim that (3.11) does not have at pole at z = 0. To see this, recall that for an n × n matrix g = (gij) having an LDU factorization, the entries of the factors can be written explicitly as ratios of determinants:

image

where imageis the determinant of the kth principal submatrix,

image; for i > j,

image (3.14)

and for i > j,

image

Apply this to image 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 g2 as in (II.3) (initially for the restriction to a small circle about 0; but because g2 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 and II.1; (3.8). To reiterate, we are now assuming that the sequence image is affine periodic.

Theorem 3.2. Suppose that image Consider the following three statements:

(I.1) image , and for each complex irreducible representation image , with lowest weight vector image has holomorphic extension to Δ, is nonzero at all z ∈ Δ, and is a positive multiple of v at z = 0.

(I.2) image has a factorization of the form

image

where g(ηj) = k(ηj) for some imagewhen image is a compact type (resp. non-compact type) root and the sequence image is rapidly decreasing.

(I.3) image has triangular factorization of the form image where image has smooth boundary values.

Then statements (I.1) and (I.3) are equivalent. (I.2) implies (I.1) and (I.3).

Moreover, in the notation of (I.2),

image

Similarly, suppose that image and image. Consider the following three statements:

(II.1) image ; and for each complex irreducible representation image, with highest weight vector imagehas holomorphic extension to Δ, is nonzero at all z∈Δ, and is a positive multiple of v at z = 0.

(II.2) image has a factorization of the form

image

where image for some ζj ∈ C (resp. g(ζj) = q(ζj) for some ζj ∈ Δ) when image is a compact type (resp. non-compact type) root and the sequenceimage is rapidly decreasing.

3. image has triangular factorization of the form image has smooth boundary values.

Then statements (II.1) and (II.3) are equivalent. (II.2) implies (II.1) and (II.3).

Also, in the notation of (II.2),

image (3.16)

Conjecture 3.2. If g1 is in the identity connected component of the sets in (I.1) and (I.3), then the converse holds, i.e. g1 has a root subgroup factorization as in (I.2). If g2 is in the identity connected component of the sets in (II.1) and (II.3), then the converse holds, i.e. g2 has a root subgroup factorization as in (II.2).

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 g2 has a triangular factorization with l2 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 imageis a matrix subgroup of image(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 image 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 image as a multiplication operator on the Hilbert space image , and we write

image

relative to the Hardy polarization image, where image is the compression of image, the subspace of functions in image with holomorphic extension to Δ. To show that g2 has a Birkhoff factorization, we must show that A(g2) is invertible (Theorem 1.1 of [11]).

Let C1,..,Cn denote the columns of image, and let imagedenote the rows of g2. We can regard these as dual bases with respect to the pairing given by matrix multiplication, i.e., image

The hypothesis of (II.1) implies that both C1 and image have holomorphic extensions to Δ (in the latter case, by considering the dual representation). Now suppose that image is in the kernel of A(g2). Then

image (3.17)

where ()+ denotes projection to image . Since image has holomorphic extension to Δ, image and therefore image is identically zero on S1 by (3.17). This implies that for image s a linear combination of the n − 1 columns Cj(z), j<n. We write

image

where the coefficients are functions on the circle (defined a.e.). Now consider the pointwise wedge product of image vectors

image

The vectors image extend holomorphically to Δ, and never vanish, for any j, by (II.1) (by considering the representation image. Since f also extends holomorphically, this implies that image has holomorphic extension to . Now

image

by (3.17) and duality.

Since the right hand side is holomorphic in Δ, by (3.17) (for j = n−1) λn−1 vanishes identically. This implies that in fact f is a (pointwise) linear combination of the first n−2 columns of image . Continuing the argument in the obvious way (by next wedging f with image to conclude that image must vanish), we conclude that f is zero. This implies that ker(A(g2)) = 0. Since image is simply connected, (A(g2) has index zero. Hence (A(g2) is invertible. This implies (II.3).

It is obvious that (II.3) implies (II.1); see (3.8). Thus (II.1) and (II.3) are equivalent.

Before showing that (II.2) implies (II.1) and (II.3), we need to explain why the C limit in (II.2) exists. We first consider the projection of the product in LK. Because imageas image the condition for the product in (II.2) to converge absolutely is that image converges absolutely. So g2 certainly represents a continuous loop.

We will now calculate the derivative formally. In this calculation, we let g2(n) denote the product up to n, and image . Then

image (3.18)

image

Because we are using an affine periodic sequence of simple reflections (with period image and so on. In general, writing imageas above, and using Proposition (4.9.5) of [15] to calculate the coadjoint action,

image (3.19)

Because image , for all image , it follows that q(n) is asymptotically n. Together with Bessel’s inequality, (3.18) implies that

image

The right hand side is comparable to image is uniformly bounded in n. Thus g2 is W1 (the L2 Sobolev space) whenever image. Higher derivatives can be similarly calculated. This shows that if w n, then image. Hence ifimagethe Frechet space of rapidly decreasing sequences, then image. Together with Proposition 1.4, this implies that the product in (II.2) converges in image

Now suppose that (II.2) holds. The map from image is continuous, with respect to the standard Frechet topologies for rapidly decreasing sequences and smooth functions. The product (3.16) is also a continuous function of image and hence is nonzero. This implies that image has a triangular factorization which is the limit of the triangular factorizations of the finite products image By Theorem 3.1 and continuity, this factorization will have the special form in (II.3). Thus (II.2) implies (II.1) and (II.3).

Remark 3.3. We now want to give a naive argument for Conjecture 3.2. Suppose that we are given g2 as in (II.1) and (II.3). Recall that l2 has values in image . We can therefore write

image (3.20)

(the use of x* for the coefficients is consistent with our notation in the SU(1,1) case (II.3) of Theorem 0.1).

As a temporary notation, let X denote the set of g2 as in (II.1) and (II.3); x* is a global linear coordinate for this space. We consider the map

image (3.21)

This map induces bijective correspondences among finite sequences image and finite sequences x*, and the maps ζ to x* and x* → ζ are given by rational maps (i.e. rational in image ); however (although it seems likely) it is not known that the limits of these rational maps actually make sense even for rapidly decreasing sequences (Appendix of [11] for the SU(2) case). We will use an inverse function argument to show that the map (3.21) has a global inverse (technically, to apply the inverse function theorem, we should consider the maps of Sobolev spaces imagewhere Xn is the Wn completion of X, but we will suppress this).

Given a variation of ζ, denoted ζ′, we can formally calculate the derivative of this map,

image

As before it is clear that this is convergent, so that (3.21) is smooth. At image this is clearly injective with closed image, so that there is a local inverse. Consider more generally a fixed image , so that image for large n. Recall that the root spaces for the image are independent and fill out image . Given a variation such that 1 image, the terms in the last sum in the derivative formula (3.22) must be zero for large n. But we know that the map (3.21) is a bijection on finite image Thus for a variation of a finite number of image which maps to zero, the variation vanishes. It is clear that the image of the derivative (3.22) is closed. The image is therefore the tangent space to X (because we know that finite variations will fill out a dense subspace of the tangent space). This implies there is a local inverse. This local inverse is determined by its values on finite x*, and hence there is a uniquely determined global inverse. This shows that (II.1) and (II.3) imply (II.2).

Finally, (3.16) follows by continuity from (3.3).

Generalization of Theorem 3.3

Theorem 3.3. Suppose image

(a) The following are equivalent:

(i) image has a triangular factorization image , where l and u have image boundary values, and satisfy the conditions image

(ii) image has a (partial root subgroup) factorization of the form

image

where image are as in (I.3) and (II.3) of Theorem 3.2, respectively.

(b) In reference to (ii) of part (a),

image(3.23)

and

image (3.24)

Remarks. Suppose that image . In this case the last condition in (i) in Theorem 3.3, that image is equivalent to the condition in Theorem 0.3 that the boundary values l21 / l11 and u21 / u22 are <1 in magnitude on S1, and part (b) specializes to the statement of Theorem 0.4.

Proof. Our strategy of proof is the following. We will first show that in part (a), (ii) implies (i). In the process we will prove part (b). We will then show that (i) implies (ii).

Suppose that we are given image as in (ii). Both image have triangular factorizations by Theorem 3.2. In the notation of Theorem 3.2,

image (3.25)

since image preserves the A factor. The basic observation is that

image (3.26)

(the inverse image in the affine extension for the identity component of loops in image), and b will have a triangular factorization which we can compute. To do this requires some care with the central extension, and this involves some preparation.

Because image is the semidirect product of image , there is an isomorphism of loop groups

image

The central extension is trivial for image , and hence there is an isomorphism

image

where the action of image is the same as the conjugation action of image is a Heisenberg extension determined by the bracket (1.3).

Given image as above, let image denote the linear triangular decomposition, where image.Then (calculating in terms of the Heisenberg extension)

image

Substituting this into (3.26) we find

image

where

image

Thus, b has a triangular factorization

image

where image,

image

and

image

Thus, from (3.25), image will have a triangular factorization imagewith

image (3.27)

Thus, (ii) implies (i) in part (a). At the same time this also implies part (b).

Now we need to show that (i) implies (ii). For this direction, there is not any need to consider the central extension, so we will no longer use tildes for group elements.

Suppose image, as in (i). At each point of the circle there exist image decompositions

image (3.28)

This is a consequence of the somewhat bizarre hypotheses in (i). Then image is the involution fixing image acts as the inverse onimage under the Hermitian type assumption.

In turn, there are Birkhoff decompositions

image

for i = 1,2. Define

image

for i = 1,2. Then

image

has triangular factorization with

image

image

and similarly

image

has triangular factorization with

image

The conclusion is somewhat miraculous. On the one hand image has values in imagebecause imageis the pointwise involution fixing image in image . On the other hand

image(3.29)

has values inimage. Therefore image has values in image . It is also clear that (3.29) is connected to the identity, and hence image and thus equals image. Hence image. Thus (i) implies (ii). This completes the proof.

Acknowledgements

The first author thanks the Provost’s Teacher-Scholar Program at California State Polytechnic University Pomona for supporting this work. The second author thanks Hermann Flaschka, whose questions motivated us to consider loops in noncompact groups. We also thank Estelle Basor for many useful conversations.

References

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  2. Caine A(2006) Compact symmetric spaces, triangular factorization, and Poisson geometry. J Lie Th 18: 273-294.
  3. Caine A, Pickrell D (2008) Homogenous Poisson structures on symmetric spaces. Int Math Res Not.
  4. Caine A, Pickrell D Noncompact groups of Hermitian symmetric type and factorization, submitted to Transformation Groups.
  5. Clancey K, Gohberg I(1981) Factorization of Matrix Functions and Singular Integral Operators. Operator Theory: Advances and Applications 3.
  6. Helgason S(1979) Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press.
  7. Kac V, Wakimoto M (1988) Modular and conformal invariance constraints in representation theory of affine algebras. Adv Math 70: 156-236.
  8. Lu JH(1999) Coordinates on Schubert cells, Kostant’s harmonic forms, and the Bruhat-Poisson structure on G/B. Transformation Groups 4: 355-374.
  9. Pickrell D(2006) The diagonal distribution for the invariant measure of a unitary type symmetric space. Transform. Groups 11: 705-724.
  10. Pickrell D (2008) Homogeneous Poisson structures on loop spaces of symmetric spaces. Geom. Methods, Appls. 4.
  11. Pickrell D (2014) Homeomorphisms of and factorization.
  12. Pickrell D, Pittmann-Polletta B (2010) Unitary loop groups and factorization. J. Lie Th. 20: 93-112.
  13. Pittmann-Polletta B(2010) Factorization of loops in unitary groupsand reduced words in affine Weyl groups. University of Arizona.
  14. Pressley A, Segal G(1986) Loop groups. Oxford University Press, New York.
  15. Widom H(1974) Asymptotic behavior of block Toeplitz matrices and determinants. Adv Math 13: 284-322.
  16. Widom H(1976) Asymptotic behavior of block Toeplitz matrices and determinants. II. Adv Math 21: 1-29.

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