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Holomorphic completions of affine Kac-Moody groups

We construct holomorphic loop groups and their associated affine Kac-Moody groups and prove that they are tame Fr\'echet manifolds. These results form the functional analytic basis for the theory of affine Kac-Moody symmetric spaces, presented first in the authors thesis. Our approach also solves completely the problem of complexification of loop groups; it allows a complete description of complex Kac-Moody groups and their non-compact real forms.

of those infinite dimensional objects are supposed to be the corresponding complex or real affine Kac-Moody groups. Thus, let us start by recalling the finite dimensional blueprint.
In this case the symmetry group is a simple complex or real Lie group. A complex simple Lie group has up to conjugation a unique compact real from. All further real forms are non-compact.
The fundamental geometric objects associated to simple Lie groups are Riemannian symmetric spaces. Recall that a Riemannian manifold (M, g) is a symmetric space if the isometry group contains for any point p ∈ M contains an element σ p such that sσ p | TpM = i Id. The construction and classification of Riemannian symmetric spaces byÉlie Cartan was a central result of Riemannian geometry in the first half of the 20'th century, opening up the way to new directions of research. Besides being a class of examples of Riemannian manifolds of fundamental importance, Riemannian symmetric spaces display numerous connections to Lie theory, mathematical physics, analysis and combinatorics. Hence there was intensive research (also begun byÉlie Cartan) devoted to the construction of a class of infinite dimensional symmetric spaces, displaying similar structural properties. From the point of view of structural properties and the numerous (in part conjectured) connections to other fields such as supergravity and M -theory, especially symmetric spaces associated to Kac-Moody groups are interesting.
Attached to any compact or complex simple Lie group, or more generally to any symmetric space, there is a spherical building [AB08]. Combinatorically a building is described as a simplicial complex or as a graph, satisfying several axioms. Geometrically the building corresponding to a symmetric space M has a clear intuitive description [BH99]: suppose the dimension k of maximal flats (maximal flat subspaces), called the rank, satisfies k ≥ 2 (otherwise the building degenerates to a set of points) and choose a point p ∈ M . The building can be seen in the tangent space T p M as follows: Consider first all k-dimensional tangent spaces to maximal flats through the point p (for a compact simple Lie group choose p = Id the identity, then T p M can be identified with the Lie algebra, maximal flats through the identity are maximal tori and the tangent spaces to maximal flats are just Abelian subalgebras of the Lie algebra g) and remove all intersection points (the so-called singular points). Then each flat is partitioned into chambers and the set of all those chambers for all flats through p corresponds exactly to the chambers of the building. All other cells of the building can be seen in the set of singular points. Chambers belonging to one flat form an apartment. Nevertheless, in this description, one does not see all apartments. To this end one has to pass to the sphere at infinity, to disenthral the construction from the dependence of a base point.
Via the geometric description of the building in the tangent space the connections between the building and polar actions become manifest: Let M be a symmetric space and p ∈ M . The isotropy representation of M is the representation of K, the group of all isometries of M fixing p on T p M : For compact Lie groups the isotropy representation coincides with the Adjoint representation. The principal orbits of the isotropy representation have the following properties: (1) The principal orbits meet the tangent spaces to the flats orthogonally and have complementary dimension. In particular, the normal spaces to the orbits are integrable. Actions with this property are called polar. Conversely, any polar action is orbit equivalent to the isotropy representation of a symmetric space [BCO03].
(2) The geometry of the principal orbits as submanifolds of Euclidean space is particularly nice and simple. They are so-called isoparametric submanifolds [PT88]. Conversely, by a theorem of Thorbergsson [Tho91], isoparametric submanifolds are principal orbits of polar representations if they are full, irreducible and the codimension is not 2.
To summarize the finite dimensional blueprint: • Polar representations correspond to symmetric spaces, isoparametric submanifolds correspond roughly to polar representations. • Chambers in buildings correspond to points in isoparametric submanifolds.
• Buildings describe the structure at infinity of symmetric spaces of noncompact type.
• The geometric realization of buildings can be equivariantly embedded via the isotropy representation into the tangent space of a symmetric space.
Let us now turn to the infinite dimensional counterpart: Affine twin buildings have been investigated by Ronan and Tits as a tool to understand minimal affine Kac-Moody groups [Ron03]. They play a similar role for affine Kac-Moody groups as classical buildings play for finite dimensional semisimple Lie groups.
Affine Kac-Moody groups are distinguished among other classes of infinite dimensional Lie groups by two important properties: • They share the most important structure properties of finite dimensional simple Lie groups, i.e. they have BN -pairs, Iwasawa and Bruhat decompositions and highest weight representations [PS86], [Kac90], [Kum02], [MP95]. The main differences can be traced back to the Weyl group being finite for Lie groups and infinite for Kac-Moody groups. • They have good explicit linear realizations in terms of 2-dimensional extensions of loop algebras and loop groups; thus they allow the use of functional analytic methods and the definition of manifold structures. The resulting Kac-Moody algebras and Kac-Moody groups are called of "analytic" type, in contrast for example to formal completions [Tit84].
For example, it is well known that affine analytic Kac-Moody groups are symmetry groups of (Kac-Moody) symmetric spaces [Fre09]. The structure theory and the classification of those spaces parallels closely the theory of finite dimensional Riemannian symmetric spaces, making them the appropriate generalization. Furthermore there is a theory of polar actions on Hilbert spaces [Ter95] and proper Fredholm isoparametric submanifolds in Hilbert spaces [Ter89]. Principal orbits of polar action on Hilbert spaces are proper Fredholm isoparametric submanifolds in Hilbert spaces. Thus a broad generalization of the finite dimensional blueprint appears. We find the same classes of objects, satisfying similar relations among each other and sharing similar structure properties [Hei06] [Fre12c]. Nevertheless, a serious gap in the infinite dimensional theory described so far is the lack of an object generalizing spherical buildings to Kac-Moody geometry. For algebraic Kac-Moody groups there are algebraic twin buildings, but they do not work well with completions as the action of any completed Kac-Moody group does not preserve the twinning; thus affine twin buildings are of restricted use in the realm of a Kac-Moody geometry. This problem can be seen also on the group level, as analytic Kac-Moody groups do not allow for the definition of a twin BN -pair or a BN -pair [AB08], [PS86]. We gave a solution to this problem introducing the notion of twin cities in [Fre11b,Fre11a].
As Kac-Moody symmetric spaces are tame Fréchet manifolds, we start in section 2 by reviewing the theory of tame Fréchet structures following the presentation given by Richard Hamilton [Ham82]; then in section 3 we introduce certain spaces of holomorphic maps and prove, that they are tame Fréchet; these spaces reappear in our construction of the tame structures on affine Kac-Moody groups. In section 4 we prove then, that holomorphic loop algebras are tame Fréchet Lie algebras and that holomorphic loop groups are tame Fréchet Lie groups. To achieve this result, we need to establish various technical results about Fréchet manifolds. In section 5 we extend the theory of polar actions from the classical Hilbert space setting [Ter95] to our setting of tame Fréchet spaces; in the final section 7 we turn our attention to holomorphic Kac-Moody groups and prove that they are tame Fréchet Lie groups.

Tame Fréchet manifolds
2.1. Fréchet spaces. In this introductory section we collect some standard results about Fréchet spaces, Fréchet manifolds and Fréchet Lie groups. Further details or omitted proofs can be found in Hamilton's article [Ham82].
Definition 2.1 (Fréchet space). A Fréchet vector space is a locally convex topological vector space which is complete, Hausdorff and metrizable.
Lemma 2.2 (Metrizable topology). A topology on a vector space is metrizable iff it can be defined by a countable collection of seminorms.
Proof. For the proof see [Ham82].
Let us look at some examples: Example 2.3 (Fréchet spaces).
(1) Every Banach space is a Fréchet space. The countable collection of norms contains just one element.
(2) Let Hol(C, C) denote the space of holomorphic functions f : C −→ C. Let furthermore K n be a sequence of simply connected compact sets in C, such that K n ⊂ K n+1 and K n = C. Let f n := sup z∈Kn |f (z)|. Then Hol(C, C; n ) is a Fréchet space.
(3) More generally, for every Riemann surface S the sheaf of holomorphic functions carries a Fréchet structure which is defined similarly as in the special case S = C discussed in the second example.
Definition 2.4. A Fréchet manifold is a (possibly infinite dimensional) manifold with charts in a Fréchet space such that the chart transition functions are smooth.
While it is possible to define Fréchet manifolds in this way, there are two strong impediments to the development of analysis and geometry of those spaces: (1) In general there is no inverse function theorem for smooth maps between Fréchet spaces. For counterexamples and examples showing features special to Fréchet spaces, see [Ham82].
(2) In general the dual space of a Fréchet space is not a Fréchet space. Dealing with those problems is difficult: let us investigate as a typical example the space of holomorphic functions Hol(C, C). C can be interpreted as a direct limit of the sets K n := B n (0) with respect to inclusion; the space of functions on a direct limit is an inverse limit; Hol(C, C) should thus be interpreted as the inverse limit of a sequence of function spaces Hol(K n , C), where By a choice of appropriate norms on the spaces Hol(K n , C), one can give them structures as Bergmann-(or Hardy-) spaces. See for example [HKK00] and [Dur00]. Hence Hol(C, C) can be interpreted as inverse limit of Hilbert spaces. By category theory, the categorical dual space of an inverse limit is a direct limit and vice versa. Thus it is clear that we cannot expect the dual space of a Fréchet space to be Fréchet. It is Fréchet iff the projective limit and the inductive limit coincide, that is, if they both stabilize. This is the case exactly for Banach spaces [Sch80].
The solution to the first problem is based on a more refined control of the projective limits. Using this structure, the inverse function theorems on the Hilbert-(resp. Banach-) spaces in the sequence piece together to give an inverse function theorem on the limit space; this is the famous Nash-Moser inverse function theorem. In the next sections we formalize those concepts.
Let us note that there are other ways to deal with these analytic problems. A recently proposed method is the concept of bounded Fréchet geometry developed by Olaf Müller [Mül06,Mül08].
The solution to the second problem consists in avoiding dual spaces. Let us remark that the theory could also be formulated by defining dual spaces as direct limit locally convex topological vector spaces.
2.2. Tame Fréchet spaces. The central problem for all further structure theory of Fréchet spaces is a better control of the set of seminorms. For a single Fréchet space this is done by the concept of a grading; much deeper is the question of comparing collections of seminorms on two different Fréchet spaces: for two Fréchet spaces F and G and a map ϕ : F −→ G this is done by imposing estimates similar in spirit to the concept of quasi isometries relating the sequences of norms ϕ(f ) n and f m , leading to the concept of "tame" maps. The property of a Fréchet space to be tame is defined then by "tame equivalence" to some model space Σ(B), which is essentially a space of holomorphic functions; the standard reference for these considerations is the article [Ham82]. We are slightly sharpening the concept of "tame equivalence" introduced by Hamilton to define (r, b, C(n))-equivalence and emphasize the nature of tame spaces as a generalization of spaces of holomorphic functions.
The prerequisite for estimating norms under maps between Fréchet spaces is a comparison of the norms on the Fréchet space itself. This leads to the notion of a grading: Definition 2.5 (grading). Let F be a Fréchet space. A grading on F is a collection of seminorms { n , n ∈ N 0 } that define the topology and satisfy Lemma 2.6 (Constructions of graded Fréchet spaces).
(1) A closed subspace of a graded Fréchet space is a graded Fréchet space.
Proof. For the proof see [Ham82].
Every Fréchet space admits a grading. Let (F, n,n∈N ) be a Fréchet space. Then F , n,n∈N such that F = F as a set and n := n i=1 i is a graded Fréchet space. The Fréchet topologies of F and F coincide. The existence of a grading is thus not a property of the Fréchet space as a metrizable topological space. It is an additional structure which is geometric in nature.
Definition 2.7 (Tame equivalence of gradings). Let F be a graded Fréchet space, r, b ∈ N and C(n), n ∈ N a sequence with values in R + . The two gradings { n } and { } are called (r, b, C(n))equivalent iff f n ≤ C(n) f n+r and f n ≤ C(n) f n+r for all n ≥ b .
They are called tame equivalent iff they are (r, b, C(n))-equivalent for some (r, b, C(n)).
The following example is basic: Example 2.8. Let B be a Banach space with norm B . Denote by Σ(B) the space of all exponentially decreasing sequences {f k }, k ∈ N 0 of elements of B. On this space, we can define different gradings: Lemma 2.9. On the space Σ(B) the two gradings f l n 1 and f l n ∞ are tame equivalent. Proof. For the proof see [Ham82].
Example 2.10. This example is used in section 3.
• The space Σ(C 2 , Eucl) of exponentially decreasing sequences of elements in C 2 where C 2 is equipped with the Euclidean norm is a tame Fréchet space. • The space Σ(C 2 , Sup) of exponentially decreasing sequences of elements in C 2 where C 2 is equipped with the supremum-norm (c 1 , c ′ 1 ) B := sup(|c 1 |, |c ′ 1 |) is a tame Fréchet space. Let us give an intuitive characterization of tame spaces Lemma 2.11. Let B be a complex Banach space. The space Σ(B) of exponentially decreasing sequences in B is isomorphic to the space Hol(C, B) of B-valued holomorphic functions.
Proof. We have to prove both inclusions: Let first f ∈ Hol(C, B) with its Taylor series expansion: The coefficients f k are elements of B and we have to show that (f k ) is an exponentially decreasing sequence. We want to use the l ∞ -norm. For the estimate of f k we need three ingredients: (1) As f (z) is entire the expansion f (z) = ∞ n=0 f k z k converges for all z ∈ C -hence sup z∈B(0,e n ) |f (z)| < ∞ of all n.
(2) Differentiation of f yields the identity: The Cauchy inequality [BG91], 2.1.20: Let f be holomorphic on B(0, r). Then Combining these ingredients we get for n ∈ N Conversely for any exponentially decreasing sequence (f n ) n∈N , f n ∈ B we define the function We have to prove that this defines an entire holomorphic function as we get for any z ∈ B(0, e n ) the estimate For fixed n the sequence (a k ) corresponds to the coefficients of the Laurent series of a holomorphic function converging on the disc B(0, e n ). For f (z) := a k z k this is proved by the estimates |a k |e kn < ∞ As this condition is satisfied by assumption for all n ∈ N we get that the function defined by the series (a k ) k∈N is holomorphic on C. Hence Σ(B) = Hol(C, C).
Let F , G, G 1 and G 2 denote graded Fréchet spaces.
Definition 2.14 (Tame linear map). A linear map ϕ : F −→ G is called (r, b, C(n))-tame if it satisfies the inequality ϕ(f ) n ≤ C(n) f n+r . ϕ is called tame iff it is (r, b, C(n))-tame for some (r, b, C(n)).
Let us give an example showing that tameness of maps depends on the grading. Let us look at the following variant of an example of [Ham82]: Example 2.15. Let P be the space of entire holomorphic functions periodic with period 2πi and bounded in each left half-plane. Define L : P −→ P by Lf (z) = f (2z).
(1) Define first the grading on P by: Then Lf n ≤ f 2n , hence L is not tame.
Definition 2.16 (Tame isomorphism). A map ϕ : F −→ G is called a tame isomorphism iff it is a linear isomorphism and ϕ and ϕ −1 are tame maps.
Definition 2.17 (Tame direct summand). F is a tame direct summand of G iff there exist tame linear maps ϕ : F −→ G and ψ : G −→ F such that ψ • ϕ : F −→ F is the identity.
Definition 2.18 (Tame Fréchet space). F is a tame Fréchet space iff there is a Banach space B such that F is a tame direct summand of Σ(B). Lemma 2.19 (Constructions of tame Fréchet spaces).
(1) A tame direct summand of a tame Fréchet space is tame.
(2) A cartesian product of two tame Fréchet spaces is tame.  [Ham82]. In section 3 we study in detail the tame Fréchet spaces of holomorphic functions which we need for the construction of Kac-Moody symmetric spaces.
Let us now introduce the notion of a tame Fréchet Lie algebra: Definition 2.20 (Tame Fréchet Lie algebra). A Fréchet Lie algebra g is a tame Lie algebra iff it is a tame vector space and ad(X) is a tame linear map for every X ∈ g.
The condition on ad(X) assures the tame structure to be invariant under the adjoint action. We clearly have the following class of examples: Example 2.21. Any finite dimensional Lie algebra is tame.
Example 2.22. The realizations of the loop algebras L(g, σ) are tame Fréchet Lie algebras for H 0 -Sobolev loops, smooth loops and holomorphic loops -compare section 6.
Up to now all maps, we studied were linear maps. Let us now proceed by a short review of some nonlinear tame Fréchet objects: Example 2.24. Suppose F and G are Banach space (hence the collection of norms consists of one norm) and Φ : F −→ G is a (r 1 , b 1 , C 1 )-tame isomorphism with a (r 2 , b 2 , C 2 )-tame inverse Φ −1 . If b 1 ≥ 2 and b 2 ≥ 2 the condition on the norms vanishes.
After some manipulations we find: which is similar to the definition of a quasi isometry [BBI01] Lemma 2.25 (Construction of tame maps).

Tame Fréchet manifolds.
Definition 2.26 (Tame Fréchet manifold). A tame Fréchet manifold is a Fréchet manifold with charts in a tame Fréchet space such that the chart transition functions are smooth tame maps.
Example 2.27. Every Banach manifold is a tame Fréchet manifold.
Definition 2.28. Let M and N be two tame Fréchet manifolds modeled on F resp. G. A map f : M −→ N is tame iff for every pair of charts ψ i : For the construction of tame structures on Kac-Moody groups we need to introduce the following new notion:  This result was proved in [Fre12f].
Lemma 2.31. Let M ⊂ F be a tame Fréchet submanifold of finite type. Let H be a tame Fréchet This means, that we chose the tame structure on a tame Fréchet submanifold of finite type compatible with the tame structure on the vector the manifold is embedded in. We need this result in the proof that affine Kac-Moody groups carry a tame structure.
Proof. Let ϕ be tame as a map ϕ F : H → F . Then the cocatenation ϕ i • ϕ is tame for any chart ϕ of M . Thus ϕ is tame as a map into M .
The technical reason for working in the category of tame Fréchet spaces and tame maps is the following Nash-Moser inverse function theorem. We cite the version of [Ham82].
Theorem 2.32 (Nash-Moser inverse function theorem). Let F and G be tame Fréchet spaces and Φ : U ⊆ F −→ G a smooth tame map. Suppose that the equation for the derivative DΦ(f )h = k has a unique solution h = V Φ(f )k for all f ∈ U and all k and that the family of inverses V Φ : U × G −→ F is a smooth tame map. Then Φ is locally invertible, and each local inverse Φ −1 is a smooth tame map.
A description of this theorem a proof and some of its applications is the subject of the article [Ham82]. In comparison to the classical Banach inverse function theorem the important additional assumption is that the invertibility of the differential is assumed not only in a single point p but in a small neighborhood U around p. This additional condition is necessary [Ham82] because in contrast to the Banach space situation it is not true in the Fréchet space case that the existence of an invertible differential in one point leads to invertibility in a neighborhood. Let us note the following result [Ham82], theorem 3.1.1. characterizing the family of smooth tame inverses.
Theorem 2.33. Let L : (U ⊆ F ) × H −→ K be a smooth tame family of linear maps. Suppose that the equation has a unique solution h for all f and k and that the family of inverses V (f )k = h is continuous and tame as a map from K toH Then V is also a smooth tame map We discuss these construction in more detail in [Fre12b].

Tame Fréchet spaces for Kac-Moody symmetric spaces
In this section we prove the tameness of certain Fréchet spaces of holomorphic functions on C * . We show first, that the space Hol(C * , C) is a tame Fréchet space; then we extend this result to the space Hol(C * , V C ), where V C denotes a complex vector space. We use these results in section 4.2 for the construction of tame Fréchet affine Kac-Moody algebras and their Kac-Moody groups via the loop algebra realization (resp. loop group realization).
In [Ham82] the following result is shown: Corollary 3.2. The space F := Hol(C, C n ) is a tame Fréchet space.
The proof of R. Hamilton makes strong use of the following observation: Our aim is now to prove that Hol(C * , C) is a tame Fréchet space. Our strategy parallels the one used by R. Hamilton for the proof of lemma 3.1 and 3.3. Let us introduce some notation: Notation 3.1.
• Let A n denote the annulus A n := {z ∈ C * |e −n ≤ |z| ≤ e n } and define the boundaries of A n by ∂A + n := {z| |z| = e n } and ∂A − n := {z| |z| = e −n }, At some points we apply the Cauchy integral formula and the maximum principle for holomorphic functions.
We start with our version of lemma 3.3. The proof of our result is considerably more involved because holomorphic functions on C * may be unbounded in the limit |z| → ∞ and |z| → 0. (1) We show: We now prove, that Hol(C * , C) is a tame Fréchet space.
Recall from definition 2.18, that we have to find a Banach space B, such that Hol(C * , C) is a tame direct summand of Σ(B). We choose B = C 2 . This Proof. Let f := k∈Z c k z k and set f + 0 := k∈N0 c k z k and f − := −k∈N c k z k . Clearly f + 0 (z) and We use the notation c k := (c k , c −k ) ⊂ C 2 and use the supremum-norm on C 2 .
(1) We show: We use the following result in the proof that an affine Kac-Moody algebra is a tame Fréchet Lie algebra (cf. definition 2.20): Lemma 3.6. The differential d dz : For the proof of lemma 3.6 we need the following technical result: Lemma 3.7. Let f be analytic on a closed disc D(z 0 , R), R > 0. Let 0 < R 1 < R. Denote by f R the supremum norm of f on the circle of radius R. Then for z ∈ D(z 0 , R 1 ), we have: Proof. This lemma is an application of Cauchy's integral formula. For details [Lan99], pp. 131. Proof of lemma 3.6.
(2) To prove tameness we use lemma 3.7. Let z ∈ A n and choose R = e −(n+1) (e − 1). Thus In the notation of lemma 3.7 we can use that R 1 = 0 and k = 1 and calculate in this way: This description is independent of z. Thus Thus the differential is (1, 0, e n+1 e−1 )-tame.
A further class of spaces that is important for the description of twisted Kac-Moody algebras (compare definition 4.6) are spaces of holomorphic functions that satisfy some functional equation. We describe first the general setting and specialize then to the two most important cases, namely symmetric and antisymmetric holomorphic functions. Proof. As usual for f ∈ Hol k,l (C * , C), we put f n := sup z∈An |f (z)|. As Hol k,l (C * , C) is a closed subspace of Hol(C * , C), it is a tame Fréchet space as a consequence of lemma 2.19.
Twisted affine Kac-Moody algebras arise as fixed point algebras of diagram automorphisms of non-twisted affine Kac-Moody algebras. The list of possible diagram automorphism shows that nontrivial diagram automorphisms have order k = 2 or k = 3 [Car02]. Thus the values of k which are important for us are k = 2 and k = 3. For k = 2, lemma 3.8 has the corollaries: Corollary 3.9 (symmetric and antisymmetric loops).
• The space Hol s (C * , Lemmata 2.19 and 3.5 and corollary 3.9 include the following result: Corollary 3.10. F := Hol(C * , C n ), F s := Hol s (C * , C n ) and F a := Hol a (C * , C n ) are tame Fréchet spaces.
Let V n be a n-dimensional complex vector space. We want a tame structure on Hol(C * , V n ). Using the corollary 3.10 we get a tame Fréchet structure on the space F := Hol(C * , C n ). This yields a tame structure on the spaces Hol(C * , V n ), Hol s (C * , V n ) and Hol a (C * , V n ) only after the choice of an identification of V n with C n , hence a choice of a basis. As this construction uses this identification of V n with C n we have to prove that the resulting tame structure is independent of it. This is the content of the following lemma: Lemma 3.11. Let V n be a n-dimensional complex vector space equipped with two norms | . | and | . | ′ . Study the spaces Hol k,l (C * , V n ). Define gradings f n := sup Corollary 3.12. Any identification of ϕ : C n −→ V n yields a 1-norm v = v i on V n . The tame structures induced on Hol(C * , C) by two different choices are equivalent. Hence the tame structure on V n is independent of the identification of V n with C n .
As a consequence we have established the following result: Theorem 3.13. Let V n be a complex vector space and any norm on V n . The space Hol(C * , V n ) with the family of norms is a tame Fréchet space.
Corollary 3.14. Let V n be a complex vector space and | | 2 the Euclidean norm on V n . The space Hol(C * , V n ) with the family of norms is a tame Fréchet space.

Tame structures on loop algebras and loop groups
In this chapter we introduce the functional analytic setting, which we use for the construction of affine Kac-Moody symmetric spaces in [Fre12a]: We define Kac-Moody groups and Kac-Moody algebras of holomorphic loops and prove that they are tame Fréchet Lie groups resp. Lie algebras. Via the loop realization Kac-Moody groups (resp. algebras) can be viewed as 2-dimensional extensions of loop groups. From a functional analytic point of view, a 2-dimensional extension is unproblematic; the only exception is, that the action of group elements on the two dimensional extension has to be well-defined as well. Nevertheless remark that this extension plays a crucial role for geometric considerations. In consequence we focus our attention on the loop group (resp. algebra). In subsection 4.2 we describe the loop algebras, in subsection 4.3 we turn our attention to the loop groups.
4.1. Affine Kac-Moody algebras. In this section we recall some basic facts about affine Kac-Moody algebras and introduce the more general notion of geometric affine Kac-Moody algebras. A thorough study of the algebraic properties of geometric affine Kac-Moody algebras is done in [Fre12d]. The loop algebra realization of (algebraic) affine Kac-Moody algebras is developed in the books [Kac90] and [Car02] from an algebraic point of view. We follow the more geometric approach to loop algebra realizations of Kac-Moody algebras, using the notion of [Ter95,HG09]. Let g be a finite dimensional reductive Lie algebra over F = R or C. Hence by definition g = g s ⊕ g a is a direct product of a semisimple Lie algebra g s with an Abelian Lie algebra g a . Let furthermore σ = σ s ⊗ σ a be some involution of g, such that σ s ∈ Aut(g s ) denotes an automorphism of finite order of g s such that the restriction of σ to any simple factor g i of g is an automorphism of g i and σ a = σ| ga = Id. If g s is a Lie algebra over R we assume g s to be a Lie algebra of compact type. We define the loop algebra L(g, σ) as follows , f satisfies some regularity conditions} . We use the notation L(g, σ) to describe in a unified way constructions that can be realized explicitly for loop algebras satisfying various regularity conditions. Regularity conditions used in applications include the following: • real analytic, denoted L an g, • (after complexification of the domain of definition) holomorphic on C * , denoted M g, • holomorphic on an annulus A n ⊂ C, denoted A n g, or • algebraic (or equivalently: with a finite Fourier expansion), denoted L alg g.
Definition 4.1 (affine Kac-Moody algebra). The indecomposable geometric affine Kac-Moody algebra associated to a pair (g, σ) as described above is the algebra: equipped with the lie bracket defined by: Here f ∈ L(g, σ) and ω is a certain antisymmetric 2-form on M g, satisfying the cocycle condition. If the regularity of L(g, σ) is chosen such that L(g, σ) contains non-differentiable functions, then d is defined on the (dense) subspace of differentiable functions.
Explicit realizations using different regularity conditions are common. We use holomorphic or algebraic loops; for these we can define Let us reformulate the definition for non twisted affine Kac-Moody algebras in terms of functions on C * : Assume σ = Id. First we develop a function f ∈ L(g, Id) into its Fourier series f (t) = a n e int . Then this function is naturally defined on a circle S 1 ; we understand this circle to be embedded as the unit circle {z ∈ C * ||z| = 1} ⊂ C * ; in this way the parameter t gets replaced by the complex parameter z := e it , with |z| = 1; understanding the Fourier expansion now as a Laurent expansion of F , we can calculate the annulus, A n , on which this series is defined. For example the holomorphic realization M g, is defined by the condition, that for any f ∈ M g the Fourier expansion describes a Laurent series expansion of a holomorphic function on C * .
Definition 4.2. The complex geometric affine Kac-Moody algebra associated to a pair (g, σ) is the algebra equipped with the lie bracket defined by: As d dt e itn = ine itn = inz n = iz d dz z n both definitions coincide. Definition 4.3 (semisimple geometric affine Kac-Moody algebra). A geometric affine Kac-Moody algebra L(g, σ) is called 4.2. Lie algebras of holomorphic maps. Let g be a complex reductive Lie algebra that is a direct product of simple Lie algebras with an Abelian Lie algebra. Simple Lie algebras are completely classified by their root systems; a complete list is given as follows: Abelian Lie algebras are classified by their dimension. A complex reductive Lie algebra g C has an up to conjugation unique compact real form g c . This compact real form is defined as the real reductive Lie algebra g R such that g is a direct product of the (up to conjugation) unique compact real forms of the simple factors of g C together with a compact real form of the Abelian factor. A compact real Abelian Lie algebra of dimension n is just R n together with the trivial bracket; but we define the exponential function such that the exponential image is a torus. Hence a compact real Lie algebra can be identified with the purely imaginary part of a complex Abelian Lie algebra.
Definition 4.4 (complex holomorphic non-twisted Loop algebra). Let g C be a finite-dimensional reductive complex Lie algebra.
(1) The loop algebra A n g C is the vector space equipped with the natural Lie bracket: (2) The loop algebra M g C is the vector space equipped with the natural Lie bracket: (1) M g C is a tame space.
(2) A n g C is a Banach space.
Proof. The first assertion is a consequence of corollary 3.10. The second assertion is a consequence of Montel's theorem stating that absolute convergent sequences of holomorphic functions converge to a holomorphic function [BG91].
The inclusions S 1 = A 0 ⊂ . . . A n ⊂ A n+1 ⊂ · · · ⊂ C * induce the reversed inclusions on the associated loop algebras: To describe the twisted loop algebras we recall the graph automorphisms of the finite dimensional simple Lie algebras: the following list contains the simple algebras A with a nontrivial diagram automorphism σ and the type of the fixed point algebra (compare [Car02]).

A
: Definition 4.6 ((twisted) loop algebra, ord(σ) = 2). Let g C be a finite dimensional semisimple complex Lie algebra of type A k , D k,k≥5 or E 6 , σ the diagram automorphism. Let g C := g 1 C ⊕ g −1 C be the decomposition into the ±-eigenspaces of σ. Let X ∈ {A n , C * }. If X = A n holomorphic functions on X are understood to be holomorphic on an open set containing X.
Then the loop algebra (Xg) σ is the vector space equipped with the natural Lie bracket: Remark 4.7 ((twisted) loop algebra, ord(σ) = 3). For the algebra of type D 4 there exists an automorphism σ of order 3. In this case we get exactly the same results as for the other types. The main difference is that we have three eigenspaces, corresponding to {ω, ω 2 , ω 3 = 1} for ω = e  (1) A n g σ equipped with the norm n is a Banach Lie algebra, (2) M g σ equipped with the norms n is a tame Fréchet Lie algebra. Proof. Closed subspaces of Banach spaces are Banach spaces and closed subspaces of tame Fréchet spaces are tame Fréchet spaces (lemma 2.19).
Theorem 4.9. The following topologies on M g C are equivalent: (1) the compact-open topology, (2) the topology of compact convergence, Then for every n 0 ∈ N and ǫ > 0 there is a k 0 such that for all k ≥ k 0 the estimate f k − f 0 n ≤ ǫ is satisfied for all n ≤ n 0 . Hence .
Hence (f k ) ⊂ M g converges in the Fréchet topology. Conversely let (f k ) ⊂ M g be a sequence converging to f 0 in the Fréchet topology. Then we have As for any compact set K ⊂ C * there is some n such that K ⊂ A n , this yields compact convergence.
The adjoint action ad(g) : M g σ −→ M g σ is (0, 0, 2 g n )-tame for each g ∈ M g σ . Contrast this with the situation for affine Kac-Moody algebras described in section 6. Having described the holomorphic complex loop algebras which we need, we turn now to some objects derived from them, namely the compact real forms and spaces of differential forms. We start with real forms of compact type: Definition 4.10 (compact real form of a holomorphic non-twisted loop algebra). Let g C be a finite-dimensional semisimple complex Lie algebra and g its compact real form. The loop algebra Xg σ R is the vector space Xg σ R := {f ∈ Xg σ C |f (S 1 ) ⊂ g} , equipped with the natural Lie bracket: As a holomorphic function on X can be expanded into its Laurent series, one can represent every element of a loop algebra by a series f (z) := n g n z n with g n ∈ g.
Lemma 4.11. The condition f (S 1 ) ⊂ g R is equivalent to the condition g n = −ḡ t −n .
Proof. Let z = e it ∈ S 1 ⊂ C * and let g n = g r n + ig i n be the decomposition of g n into its real and imaginary parts. Then we find f (z) = n∈Z g n z n = n∈Z g n e itn = = a 0 + n∈N g n e itn + g −n e −itn = = a 0 + n∈N (g r n + ig i n )(cos(tn) + i sin(tn)) + (g r −n + ig i −n )(cos(−tn) + i sin(−tn)) = = a 0 + n∈N (g r n cos(tn) − g i n sin(tn)) + i(g i n cos(tn) + g r n sin(tn))+ (g r −n cos(−tn) − g i −n sin(−tn) + i(g i −n cos(−tn) + (g r −n sin(−tn)) = = a 0 + n∈N (g r n + g r −n + i(g i n + g i −n )) cos(tn) + (−g i n + g i −n + i(g r n − g r −n )) sin(tn) This gives for the coefficients g r n : In a similar way we get for the imaginary parts of the coefficients g i n = (g i −n ) t and thus the result.
Lemma 4.12. M g R is a tame Fréchet space.
Proof. M g R ⊂ M g C is a closed subspace and thus tame according to lemma 2.19. Definition 4.13.
(1) Ω 1 (X, g C ) is the space of g C -valued 1-forms on X; elements ω ∈ Ω 1 (X, g C ) are of the form ω(z) = f (z)dz with f (z) ∈ Xg C . We define a family of norms by ω n := |f (z)| n . (2) Ω 1 (X, g C ) R is the space of g C -valued 1-forms on X such that f (S 1 ) ⊂ g R .
As M g C and M g R are tame Fréchet spaces, also Ω 1 (X, g C ) and Ω 1 (X, g C ) R are tame. Remark, that this is not the topology as a dual space.
Remark 4.14. Real forms of the algebras Xg σ C correspond to conjugate-linear involutions of Xg σ C : assign to a real form the conjugation with respect to it. In the other direction, fixed point algebras of conjugate-linear involutions are real forms. Hence, real forms are closed subalgebras. Thus by an application of lemma 2.19 real forms of M g σ C are tame, real forms of A n g σ C are Banach. 4.3. Lie groups of holomorphic maps. Up to now we studied analytic structures on loop algebras but not on the associated loop groups. In short the main result is that all loop algebras interesting to us are tame Lie algebras. In this section we prove similar results for loop groups. Let G be a compact semisimple Lie group and G C its complexification.  In this section we show that the following groups are smooth tame Lie groups: Definition 4.16 (complex loop groups).
(1) The loop group A n G is the group The multiplication is defined to be f g(z) := f (z)g(z) for f, g ∈ A n G.
(2) The loop group M G is the group The multiplication is defined to be (f g)(z) := f (z)g(z) for f, g ∈ M G. Definition 4.17 (real form of the compact type).
(1) The real form of the compact type A n G R is defined to be (2) The real form of the compact type M G R is defined to be There are exponential functions Let us remark, that for any z ∈ C * resp. z ∈ A n , the curve γ(t) : The next important object needed to describe the connection between the loop algebras and the loop groups is the definition of the Adjoint action Ad: As usual it is defined pointwise using the Adjoint action of the Lie group G K , K ∈ {R, C}: . For the Adjoint action of groups of the compact type to be well-defined we have to check, that the condition f (S 1 ) ⊂ g R is preserved. This is a consequence of the adjoint action for finite dimensional compact Lie groups: for all z ∈ S 1 we have f (z) ∈ G R and h(z) ∈ g R . Thus the condition Ad(f )h(z) ∈ g R is preserved pointwise.
Lemma 4.18. The exponential function and the Adjoint action satisfy the identity: Proof. Applying the well-known identity for finite dimensional Lie algebras (resp. Lie groups) we get that the identity is valid pointwise.
We now investigate the functional analytic nature of the groups A n G and M G: to fix some notation let Xσ denote by abuse of notation an involution of Xg resp. of XG. Let XG D denote a real form of non-compact type of XG, and denote by Fix(Xσ) the fixed point group of an involution Xσ.
E. Heintze and C. Groß [HG09] show that real forms of the non-compact type of a complex simple Kac-Moody algebra are in bijection with involutions of the compact real form (which is unique up to conjugation). Let Xg R be a compact real form with involution Xσ. We denote by Xg D,σ the real form of non-compact type associated to Xσ.
Let us focus on a description of the groups A n G: as A n is compact we can follow the classical strategy to define manifold and Lie group structures. We start by defining a chart on an open set containing the identity with values in the Lie algebra via the exponential map; then we use left translation to construct an atlas of the whole group. This strategy yields the following basic results: Theorem 4.19.
(1) A n G R and A n G C are Banach-Lie groups.
(2) Real forms A n G D of non-compact type of A n G C are Banach-Lie groups.
(3) Quotients A n G R /Fix(A n σ) and A n G D,σ /Fix(A n σ) are Banach manifolds.
For Banach-Lie groups and Banach manifolds, there exists a huge literature; for a classical introduction see for example [Pal68]. For the groups M G themselves the theory is considerably more difficult. The crucial observation is the fact that the exponential map has in general not to be a local diffeomorphism. We give an example of this strange phenomenon: Example 4.20 (M SL(2, C)). We study the Lie group SL(2, C). As is well known, exp : sl(2, C) −→ SL(2, C) is not surjective. For example elements g ∈ SL(2, C) conjugate to the element −1 1 0 −1 are not in the image of exp (sl(2, C)) (see [DK00]).
We want to show that there exists a sequence f n ∈ M SL(2, C) which converges to the identity element Id In consequence f n is not contained in ℑ(Mexp(sl(2, C))) for any n ∈ N. On the other hand, for any fixed z 0 ∈ C * we find the limit lim n→∞ f n (z 0 ) = 1 0 0 1 = Id .
Hence in the compact-open topology for every neighborhood U k of the identity there exist n k ∈ N such that ∀n ≥ n k : f n ∈ U k . This oberservation concludes the proof that f n is not a local diffeomorphism.
Especially this observation contains the corollary that the groups M G are no locally exponential Lie groups in the sense of Karl-Hermann Neeb [Nee06], that is Lie groups such that exp is a local diffeomorphism. Hence, we have to find another way to define manifold structures on M G. We start by describing some results about the relationship between M G and M g. Then we show that loop groups satisfy the weaker axioms for pairs of exponential type introduced by Hideki Omori.
Definition 4.21. The tangential space T p (M G) is defined as the space of path-equivalence classes of smooth paths. Hence two curves A path depends holomorphically on the second factor and smoothly on the first factor. Let us remark, that one can use weaker regularity conditions on the first factor (i.e. C k -dependence); we will not pursue the study of these weaker regularity conditions any further. In contrast the holomorphic dependence on the second factor is derived from the regularity condition imposed on the (holomorphic) Kac-Moody group. The relationship between M g and M G is described by the following three results (we postpone the proofs after the discussion):

Moreover the tangential space T e (M G) is isomorphic to the Lie algebra of left-invariant vector fields on M G.
While the tangential space of a loop group is the corresponding loop algebra, the exponential map behaves in general badly. More precisely, we have the following result: Theorem 4.23 (Loop groups whose exponential map is no local diffeomorphism). Let G C be a complex semisimple Lie group.
Mexp : M g −→ M G is not a local diffeomorphism. This is in sharp contrast to the case of nilpotent Lie groups For these groups we have the following result: Proposition 4.24 (Loop groups whose exponential map is a local diffeomorphism). Let G C be a complex Lie group such that its universal cover is biholomorphically equivalent to C n . Then its exponential map Mexp is a local diffeomorphism.
Corollary 4.25. Let G be a complex Lie group. If G is nilpotent (i.e. Abelian) then Mexp is a diffeomorphism.
Corollary 4.25 is a direct consequence of proposition 4.24, as Abelian and nilpotent Lie groups have C n as their universal cover [Kna96], corollary 1.103 and [Var84], section 3.6.
Theorems 4.22 and 4.23 may seem contradictory at first, as they state, that Lie group −→ Lie algebra : Good behavior! Lie algebra −→ Lie group : Bad behavior! Nevertheless, it is a typical for infinite dimensional systems in the following sense: The step from the Lie group to the Lie algebra is linearization or differentiation; differentiation exists in very general frameworks; typically no subtle obstacles arise; in contrast the step from the Lie algebra to the Lie group is integration; integration is in general frameworks a very subtle procedure [KM97,BGN04,Ber08,Gro97]. This behavior seems contradictory as the usual intuition tells us there should be a geodesic in every direction. From the point of view of classical analysis, one can describe the analytic basics of this situation as a consequence of a change of limits, arising because of the non-compactness of C * : • On the one hand, theorem 4.22 describes locally uniform convergence on each compact subset K ⊂ C * . • Theorem 4.23 shows in contrast that globally on C * convergence needs no longer be uniform. We now give the proofs that we have omitted: Proof of theorem 4.22.
(1) We prove first the inclusion M g ⊂ T e (M G). Hence we have to show the following: let γ(t) : (−ǫ, ǫ) → M G be a curve in M G such that γ(0) = e ∈ M G. Thenγ(0) is in M g.
Using, that M g is a loop algebra, we have an equivalent description ofγ(0) as a functioṅ γ(0, z : C * ) −→ g, describing explicitly the loop. Let z 0 ∈ C * arbitrary but fixed. Then γ z0 (t), the evaluation of γ(t) at z 0 ∈ C * , is a curve in G satisfying γ z0 (0) = e. Thus, using that the finite dimensional exponential map exp : g −→ G is a local diffeomorphism, we get that d dt γ z0 (t) is an element in g. As z 0 is arbitrary we have constructed a function w : C * −→ g, satisfying w(z) =γ(0, z) ; we have now to prove that w is holomorphic.
We use, that γ(t, z) depends holomorphically on z. This is equivalent to d dz γ(t, z) = 0 .  The image of the exponential map consists of those loops whose image is completely contained in the image of the Lie group exponential map. We give further comments about this situation in the third part, dealing with geometric aspects [Fre12a], where we study the behavior of geodesics.  The idea of the proof is to use logarithmic derivatives. The idea to use logarithmic derivatives to get charts is quite common: for regular Lie groups it is developed in the book [KM97], chapters 38 and 40. Furthermore it is used by Karl-Hermann Neeb [Nee06] to prove the following theorem [Nee06], theorem III.1.9.: Theorem 4.27 (Neeb). Let F ∈ {R, C}, G be a F-Lie group and M a finite dimensional, connected σ-compact F-manifold. We endow the group C ∞ F (M, G) with the compact open C ∞ -topology, turning it into a topological group. This topology is compatible with a Lie group structure if dim F M = 1 and π 1 (M ) is finitely generated.
The main ingredients for the proof of Neebs theorem are the use of logarithmic derivatives to define charts in Ω 1 (M, g), the space of g-valued 1-forms on M and Glöckners inverse function theorem [Glö07], to take care of the monodromy if π 1 (M ) is nontrivial.
If F = C we have the equivalence C ∞ F (M, G) ≃ Hol(M, G). Our situation is the special case M = C * . Hence, π 1 (M ) = Z is a finitely generated group. The compact open C ∞ -topology coincides for holomorphic maps with the topology of compact convergence. Thus Neeb's theorem tells us that M G F , F ∈ {R, C} are locally convex topological Lie groups.
Nevertheless, we do not get tame structures. Hence we have to prove the theorem completely new. Our presentation follows the proof of Karl-Hermann Neeb for the locally convex case.
The first step of the proof is the following lemma, whose statement and proof can be found in [KM97] and [Nee06]: morally it is a straight forward application of the monodromy principle for holomorphic Pfaffian systems, as described in the article [NY02].
Compare this embedding with the description of polar actions on Fréchet spaces in section 5. Let π 1 and π 2 denote the projections: We construct charts for M G C ⊂ Ω 1 (C * , g C ) × G C as direct product of charts for π 1 • ϕ(M G C ) and π 2 • ϕ(M G C ).
• π 2 • ϕ is surjective; hence a describing of charts for the second factor can be done by choosing charts for G. Via the exponential mapping and left translation, we get charts ψ 2,g : To describe the family of norms, we use the Euclidean metric n := Eucl . • The first factor is more difficult to deal with as π 1 •ϕ is not surjective. While every g C -valued 1-form α ∈ Ω(C * , g C ) is locally integrable by lemma 4.28, the monodromy may prevent global integrability. A form α ∈ Ω 1 (C * , g C ) is in the image of π 1 • ϕ iff its monodromy vanishes, that is iff e S 1 α = e ∈ G C . This is equivalent to the condition S 1 α = a −1 (α) ⊂ 1 2πi exp −1 (e) where a −1 (α) denotes the (−1)-Laurent coefficient of the Laurent series of α = f (z)dz. Hence we get the characterization of ℑ(π 1 • ϕ) as the inverse image of e ∈ G C via the monodromy map.
Thus we have to show that this inverse image is a tame Fréchet manifold. To this end, we use composition with a chart ψ : U −→ V for e ∈ U ⊂ G with values in G C . This gives us a tame map Ω(C * , g C ) −→ g C . This map satisfies the assumptions of the following theorem, whose proof can be found in [Fre12f].
Theorem 4.29. Let F be a tame space and ϕ : F → R n a tame map. Let g ∈ R n be a regular value for ϕ. Then ϕ −1 (g) is a tame Fréchet submanifold of finite type.
Thus its inverse image is a tame Fréchet submanifold. This proves that π 1 • ϕ is a tame Fréchet submanifold. Thus M G C as a product of a tame Fréchet manifold with a Lie group is a tame Fréchet manifold. This completes the proof of theorem 4.26.
Theorem 4.30. M G C is a tame Fréchet Lie group.
Proof. From theorem 4.26 we know that M G C is tame Fréchet manifold. Hence we have to check that we get for ψ 1 the description (1)) .
which is smooth as a direct product of smooth tame maps.
Similarly we get for ψ 2 : which is again a smooth tame map.
We now investigate different classes of quotients of loop groups: Proposition 4.31. M G R is a tame Fréchet Lie group.
Proof. The proof is similar to the proof for M G C . We have only to take care of the reality condition f (S 1 ) ⊂ G R for loops f ∈ M G C . Thus the embedding ϕ maps a loop f into Ω (C * , g R ) × G R , which are both tame Fréchet spaces. Now similar arguments apply.
Proposition 4.32. The group M G C /M G R is a tame Fréchet manifold.
Proof. Review the embedding 1)). This is the well-known gauge-action of the group M G R , denoted by G * (M G R ). Thus there is a well defined embedding No we study again the projections π 1 and π 2 on the first and second factor. π 2 is surjective; G C /G R is a tame manifold; so this factor is no problem.
The projection on the first factor, π 1 , needs a more careful analysis: the right multiplication of M G R on M G R is surjective: using the decomposition g C = g R + ig R , we get for Ω 1 (C * , g C ) := Ω 1 R (C * , g C ) + iΩ 1 R (C * , g C ). The surjectivity of the right multiplication of M G R on M G R translates into the surjectivity of the M G-gauge action on the imaginary part iΩ 1 R (C * , g C ) ∩ ℑ(π 1 • ψ)(M G). Thus we can suppose to have chosen a representative f ∈ f · M G, such that the imaginary part π 1 • ψ(f ) is 0. So all we have to check is the real part. Here we find that exp −1 (e) = 0. Thus a −1 = 0. So we can identify the image π 1 • ψ(M G C /M G R ) ≃ Ω 1 R (C * , g|a −1 = 0). This is a tame Fréchet space. Hence proposition 4.32 is proven.
Having proved that M G C and M G C /M G R are tame Fréchet manifolds we have to check that the same is true for the quotients M G R /Fix(ρ) and M G D /Fix(ρ).
To this end, let M G be a loop group and M ρ the loop part of an involution of the second kind. Proof. To generalize the proofs of the non-twisted setting to the twisted setting one has to check that the subspaces defined by diagram automorphism are preserved by the logarithmic derivative.
(1) For the exponential map, we use σ g resp. σ G to denote the realization of the diagram automorphism σ on g resp. G. Any involution of a semisimple Lie group satisfies the identity: (2) For the logarithmic derivative we calculate: Thus we get charts in the σ-invariant subalgebra of Ω 1 (C * , g C ).
The following definition is due to Omori [Omo97]: Definition 4.36 (Exponential pair). A pair (G, g) consisting of a Fréchet group G and a Fréchet space g is called a topological group of exponential type if there is a continuous mapping: such that: (1) For every X ∈ g, exp(sX) is a one-parameter subgroup of G.
(3) For a sequence {X n } ∈ g, lim n→∞ X n converges to an element X ∈ g iff lim n→∞ (exp sX n ) converges uniformly on each compact interval to the element exp(sX). (4) There is a continuous mapping Ad : G × g −→ g with h exp(sX)h −1 = exp sAd(h)X for every h ∈ G and X ∈ g.
Proposition 4.37 (Exponential type). The pair (M G K , M g K ) is of exponential type.
Proof. We proved that M G is a tame Fréchet Lie group, thus a topological group. To prove that (M G K , M g K ) is of exponential type, we have to check the four conditions given in definition 4.36: (1) The first condition can be checked by a pointwise analysis: for f ∈ M g and for every z ∈ C * , the curve exp sf (z) is a 1-parameter subgroup in G. This pieces together for all z ∈ C * , to yield the condition.
(2) The second condition follows analogously: let X, Y ∈ M g. X = Y iff X(z) = Y (z) for all z ∈ C * . The finite dimensional theory tells us that this is equivalent to the curves exp(sX(z)) ⊂ G K and exp(sY (z)) ⊂ G K to be equivalent for all z ∈ C * , but this is equivalent to exp(sX) = exp(sY ). This assertion is correct as for every z ∈ K : exp lim n→∞ (exp sX n (z)) = exp(sX)(z).
(4) The last assertion follows again from pointwise consideration and the validity of the assertion for finite semisimple Lie groups.
We give some remarks about 1-parameter subgroups.
Remark 4.38. Let g(t) := X exp(tu) for u ∈ Xg K be a 1-parameter subgroup in XG K , X ∈ {A n , C * }, K ∈ {R, C}. Then the following statements hold: (1) X exp(tu) z0 is a 1-parameter group in G C for all z 0 ∈ X.
(2) If A n ⊂ A n+k then the embedding A n+k G ֒→ A n G maps 1-parameter subgroups onto 1-parameter subgroups.
Remark 4.39. As we have seen, the fact that the exponential function does not define a local diffeomorphism is responsible for several difficulties; so it is reasonable to try to use a setting in which the exponential function defines a local diffeomorphism. So let us try to take loops f : S 1 −→ G satisfying some regularity condition. In this case the exponential map defines always a local diffeomorphism, as a neighborhood of the identity element of such a loop group is given by loops whose images lie in a small neighborhood V of the identity of the subjacent Lie group; this neighborhood can be chosen in a way such that the group exponential is a diffeomorphism from an open neighborhood U in the Lie algebra onto it. But now other problems appear: (1) Suppose the functions to be H 1 -Sobolev loops. In this setting, one can construct weak Hilbert symmetric spaces of compact and non-compact type. Nevertheless, one cannot define the double extension corresponding to the c-and d-part of the Kac-Moody algebra. As this extension is responsible for the structure theory, this setting is not useful for us. (2) To be able to construct the extension corresponding to the derivative d, one needs loops that are C ∞ . For C ∞ -loops, it is possible to construct compact type symmetric spaces corresponding to the finite dimensional types I and III, but for Kac-Moody symmetric spaces of C ∞ -loops there is no dualization: as the complexification of c is not defined, we cannot complexify. As a consequence there are no symmetric spaces of the non-compact type. The details of both theories are developed in [Pop05]. Summarizing we have the following observation: Proposition 4.40. The setting of holomorphic loops is the biggest setting such that Kac-Moody symmetric space of the compact type and of the non-compact type of the same regularity condition can be defined.
Let us mention in this context the short summary in [Ber03] about infinite dimensional differential geometry, where the conflict between often desirable structure as a Hilbert space and useful metrics is addressed.

Polar actions on tame Fréchet spaces
In this section we study polar representations on tame Fréchet vector space. The aim is, to show, that the adjoint representation of Kac-Moody groups of holomorphic maps M G and their associated s-representations induce polar actions; from the point of view of Kac-Moody geometry, this step assures, that there is a well-defined connection between the (local) classes of objects as are polar actions and isoparametric submanifolds and the global classes of objects i.e. affine Kac-Moody symmetric spaces and twin cities.
Let us start by sketching the finite dimensional blueprint: Let M = G/K be a finite dimensional Riemannian symmetric space. The group K stabilizes the point p = eK ∈ M ; it is called the isotropy subgroup. The linear representation K ≡ K p : T p M −→ T p M is called the isotropy representation of M . Recall furthermore that a representation K : V −→ V of a Lie group K on a vector space V is polar, if there is a subspace Σ ⊂ V , called a section, that intersects each orbit orthogonally. It is a by now classical result, that the isotropy representation of a (finite dimensional) Riemann symmetric space is a polar representation of the isotropy group on the tangential space. For the isotropy representation of a symmetric space, sections correspond to any maximal flat subspace, containing p. We prove in a subsequent part of our program describing the geometry of Kac-Moody symmetric spaces, dealing with the geometric properties of these spaces, that Kac-Moody symmetric spaces behave in a similar way. Remark nevertheless a striking difference between the finite dimensional (spherical) and the infinite dimensional affine situation: while in the finite dimensional situation all orbits have finite codimenions, ranging from 2 to the rank, in the affine case there are orbits with (small) finite codimension bounded by rk(G) and orbits with infinite codimension. This phenomenon is related to the dichothomy of complex affine Kac-Moody groups, that can be viewed either as (finite dimensional) (algebraic) groups over the ring of holomorphic functions or as (infinite dimensional) affine Kac-Moody group over the base field C (resp. R. Via duality constructions this dichothomy is also the algebraic structure behind the differences in the geometry of coadjoint orbits of Kac-Moody groups and loop groups, which is sketched for example in the book [KW09]; we gave a building theoretic interpretation in [Fre11b], relating the finite dimensional orbits to the affine twin city and the infinite dimensional ones to the spherical building at infinity [Fre12e].
We prove in this section, that the orbits with finite codimension correspond to the gauge actions of tame loop groups on tame spaces; these actions are restrictions of the adjoint action of the Kac-Moody group to certain subspaces.
Closely related is the theory of polar actions on Hilbert spaces, which is described in the article [HPTT95,PT90,Ter95,Gro00].
Important special cases are the following: let ∆ σ ⊂ G × G denote the σ-twisted diagonal subgroup of G × G, that is: (g, h) ∈ ∆ σ iff h = σ(g). We use the notation ∆ = ∆ Id for the non-twisted subgroup.
( H 0 (S 1 , g)-is polar [PT88]. In this section we describe a similar theory for the loop groups XG σ on the tame loop algebras Xg σ . As usual let X ∈ {A n , C * }. Holomorphic functions on A n are supposed to be holomorphic in an open set containing A n . From the embedding XG σ ֒→ H 1 ([0, 1], G) and Xg σ ֒→ H 0 ([0, 1], g) it is clear that the algebraic part of the theory works exactly the same in all regularity conditions. This means for example: sections for holomorphic actions correspond to sections for the Hilbert actions and the associated affine Weyl groups are the same. Hence the crucial point is to check that the additional regularity restrictions fit together. This breaks down to two points: (1) One has to show locally that the additional regularity conditions are satisfied.
(2) One has to show globally that monodromy conditions are satisfied.
To define orthogonality on Xg we use the H 0 -scalar product induced on M g by the embedding into H 0 ([0, 1], g). Hence we can define polar actions on Banach (resp. tame) spaces like that: Definition 5.2. An action of a Lie group G on a Fréchet space F is called polar iff there is a subspace S, called a section, intersecting each orbit orthogonally with respect to some scalar product.
Theorem 5.3. The gauge action of XG σ R on Xg σ is polar; an Abelian subalgebra a ⊂ g interpreted as constant loops is a section.
The proof consists of two parts: (1) We have to show that each orbit intersects the section a.
(2) We have to show that the intersection is orthogonal. The second part follows trivially from the embedding and Terng's result. Thus we are left with proving the first assertion. We do this in a step-by-step way: first we study the action of C k -loop groups on C k−1 -loop algebras (k ∈ {N, ∞}). Then we proceed to the holomorphic setting of theorem 5.3.
Lemma 5.4. The gauge action of L k G σ on L k−1 g σ is polar for k ∈ {N, ∞}.
This result is used without proof in [Pop05] in order to show that all finite dimensional flats are conjugate. We do not know if a proof can be found in the literature. For completeness we give one: Proof of lemma 5.4.
(1) Orthogonality in L k−1 g σ is defined via the embedding into the space H 0 ([0, 1], g) and the use of the H 0 -scalar product. Hence orthogonality of the intersection between sections and orbits is covered by Terng's result.
(3) The periodicity relation: We want to show that L k G σ acts on L k−1 g σ with slice a for k ∈ {N, ∞}.
(a) Let first g ∈ L k G σ and u ∈ L k−1 g σ . Then g · u = gug −1 − g ′ g −1 is in L k−1 g σ . Thus L k G σ acts on L k−1 g σ . (b) We have to show that any L k G σ -orbit intersects the section a. This is equivalent to: For each u ∈ L k−1 g σ , there is X ∈ g and g ∈ P k (G, ∆) such that exp(tX) = g(t)h(t)g −1 (0) with h ′ (t) = u(t)h(t) and the derivatives coincide; interpret in this last equation u(t) as a quasi-periodic function on R (i.e. u(t + 2π) = σu(t) and h(t) as a function on R).
After rearranging, this is equivalent to the identity σ(g −1 0 ) exp(2πX)g 0 = σ(h(t) −1 )h(t + 2π) . As the left side is a constant we find: which is the desired periodicity condition.
For n = 0 we use induction. If g is k-times differentiable then u is k − 1-times differentiable. This proves the lemma.
Proof of theorem 5.3. To prove the theorem, we have to further strengthen the used regularity conditions to holomorphic functions. The description in the proof of lemma 5.4 shows that the group of analytic loops L an (S 1 , G) acts polarly with section a on the algebra L an (S 1 , g) of analytic loops.
(1) The case of holomorphic loops on C * For the specialization to holomorphic maps we use the description: Identifying it ↔ t in this (resp. the above) description we get an embedding: This shows that there are no problems concerning the monodromy. So we have only to check the regularity aspect. For g ∈ M G and u ∈ M g, g · (u) = gug −1 − g ′ g −1 ∈ M u. On the other hand, using the description in the proof of lemma 5.4, we get for u ∈ M g a transformation function g(t) := exp(tX)g(0)h(t) −1 . A priori this function is in L(S 1 , G); but exp(tX) can be continued to a holomorphic function on C, g(0) is a constant and h(t) −1 is a solution of the differential equation: h ′ (t) = u(t)h(t); if u(t) is defined on C * , this equation has a solution on the universal cover of C * , that is C. So g(t) is defined on C, but has perhaps nontrivial monodromy; this is, of course, not possible, as the embedding tells us that g(t) ⊂ L an G.
(2) The case of holomorphic loops on A n This case is exactly similar. C is replaced by A ′ (compare subsection 3.1).
Hence theorem 5.3 is proved.
Thus we have proven that σ-actions and diagonal actions are polar. Those two cases corresponds to the isotropy representation of Kac-Moody symmetric spaces of types II and IV : the diagonal action is induced by the isotropy representation of Kac-Moody symmetric spaces in the non-twisted case, the σ-action is induced by the isotropy representation of σ-twisted one.
The isotropy representations of Kac-Moody symmetric spaces of type I and III correspond to the Hermann examples. A holomorphic version of the Hermann examples [HPTT95] can be defined in exactly the same way: Let XG σ R be a simply connected loop group, ρ an involution such that Xg σ R = K ⊕ P is the decomposition into the ±1-eigenspaces of the involution induced by ρ on Xg σ R . Let XK R ⊂ XG σ R be the subgroup fixed by ρ.
Theorem 5.5. The gauge action of XK R on P is polar.
Proof. The proof is like the one of theorem 5.3. One starts with a similar result for polar actions on Hilbert action [Ter95] and checks then step by step that the introduced higher regularity conditions fit together.

Tame structures on Kac-Moody algebras
In this section we describe explicit realizations as central extensions of holomorphic loop algebras of the abstract affine geometric Kac-Moody algebras which we introduced in definition 4.1. Let X ∈ {A n , C * }. As usual, holomorphic functions on A n are understood to be holomorphic in an open set containing A n .
Definition 6.1 (holomorphic affine geometric Kac-Moody algebra). Define Xg to be an explicit realization of L(g, σ) with L(g, σ) in the category of holomorphic maps Xg σ .
Thus an element of a Kac-Moody algebra can be represented by a triple (f (z), r c , r d ), where f (z) denotes a g C -valued holomorphic function on X and {r c , r d } ∈ C.
We equip those algebras with the norms (f, r c , r d ) n := sup z∈An |f z | + (r crc + r drd ) 1 2 . Thus we use the supremum norm on the loop algebra and complete it with an Euclidean norm on the double extension defined by c and d. Lemma 6.2 (Banach-and Fréchet structures on Kac-Moody algebras).
(1) For each n, the algebras A n g σ R and A n g σ C equipped with the norm n are Banach-Lie algebras, (2) M g σ R and M g σ C equipped with the sequence of norms n are tame Fréchet-Lie algebras.
(1) As a consequence of lemma 4.5, A n g σ is a Banach space. Thus A n g σ is Banach.
Continuity follows from the continuity of d dz , which is a consequence of the Cauchy-inequality and the boundedness of multiplication on compact domains.
(2) M g σ is a tame Fréchet space as a consequence of lemma 4.5. Thus M g σ is tame as a direct product of tame spaces (lemma 2.19). To prove tameness of the adjoint action, we need tame estimates for the norms. Those estimates follow directly from the Banach space situation: Thus ad( g) is (1, 0, 6πe 2n+1 g n+1 )-tame.
This result shows that the tame structure on the Kac-Moody algebra is preserved by the adjoint action. For additional analytic details and the Cauchy-inequalities see for example [BG91].
7. Affine Kac-Moody groups 7.1. The loop group construction of affine Kac-Moody groups. In this section we describe the construction of affine Kac-Moody groups as 2-dimensional extensions of loop groups. Our presentation of the central extensions follows a construction proposed [PS86] for a Hilbert space setting; we use the tame Fréchet setting, developed in section 4. Furthermore using a technical result of B. Popescu we prove that Kac-Moody groups of holomorphic loops carry a structure as tame Fréchet manifolds.
Let G C denote a complex semisimple Lie group and G its compact real form. As the constructions are valid in Kac-Moody groups defined with respect to various different regularity conditions, we use the regularity-independent notation L(G C , σ) for the complex loop group and L(G, σ) for its real form of compact type. To define groups of polynomial or analytic loops, we use the fact that every compact Lie group is isomorphic to a subgroup of some unitary group. Hence we can identify it with a matrix group. Similarly the complexification can be identified with a subgroup of some general linear group [PS86]. Groups of polynomial loops are then defined with respect to this representation. Kac-Moody groups are constructed in two steps.
(1) The first step consists in the construction of an S 1 -bundle in the real case (resp. a C *bundle in the complex case) that corresponds via the exponential map to the central term Rc (resp. Cc) of the Kac-Moody algebra.
(2) In the second step we construct a semidirect product with S 1 (resp. C * ). This corresponds via the exponential map to the Rd-(resp. Cd-) term Study first the extension of L(G, σ) with the short exact sequence: There are various groups X that fit into this sequence. We need to define L(G, σ) in a way that its tangential Lie algebra at the identity e ∈ L(G, σ) is isomorphic to L(g, σ).
If G is simply connected it can be shown that this object is a well defined group independent of arbitrary choices made in the construction iff ω is integral. This condition is satisfied by our definition of ω [PS86], theorem 4.4.1. If G is not simply connected, the situation is a little more complicated: let G = H/Z where H is a simply connected Lie group and Z = π 1 (G). Let (LG) 0 denote the identity component of LG. We can describe the extension using the short exact sequence: [PS86], section 4.6. . In case of complex loop groups, the S 1 -bundle is replaced by a C * -bundle.
Hence, we can now give the definition of Kac-Moody groups: Definition 7.1 (Kac-Moody group).
(1) The real Kac-Moody group M G R is the semidirect product of S 1 with the S 1 -bundle M G R .
(2) The complex Kac-Moody group M G C is its complexification: a semidirect product of C * with M G C -bundle over M G. The action of the semidirect S 1 (resp. C * )-factor is in both cases defined by a shift of the argument: Remark 7.2. Remark that in the compact case the shift is by elements w d = e iϕ d only. Hence the action preserves the unit circle S 1 . Thus one can use function spaces on S 1 thus yielding more general Kac-Moody groups than the groups of holomorphic loops, we study. Nevertheless those groups have no complexification in the same regularity class.
The next aim is to prove that Kac-Moody groups are tame Fréchet manifolds. To this end we use a result of B. Popescu [Pop05] stating that fiber bundles whose fiber is a Banach space over tame Fréchet manifolds are tame.
We start with the definition of tame fiber bundles: Definition 7.3 (tame Fréchet fiber bundle). A fiber bundle P over M with fiber G is a tame Fréchet manifold P together with a projection map π : P −→ M satisfying the following condition: For each point x ∈ M there is a chart ϕ : U −→ V ⊂ F with values in a tame Fréchet space F such that there is a chart ϕ : π −1 (U ) −→ G × U ⊂ G × F such that the projection π corresponds to a projection of U × F onto U in each fiber.
The following lemma is proved in [Pop05].
Lemma 7.4. Let P be a fiber bundle over M whose fiber is a Banach manifold; then P is a tame Fréchet manifold.
This result contains the important corollary: Corollary 7.5. The Kac-Moody groups M G R , M G C , the quotient spaces M G C / M G R , M G D and the quotient spaces M G R /Fix(ρ) and M G D /Fix(ρ) are tame Fréchet manifolds.
Next we prove: Theorem 7.6. M G C / M G R is diffeomorphic to a vector space.
Proof. By theorem 4.32, we know that M G C /M G R is diffeomorphic to a vector space. As M G R is a subgroup of M G C , the quotient is well defined. To prove the theorem we check the decomposition To this end we use the description of the elements in M G C / M G as 4-tuples. Two 4-tuples (g(z), p(z, t), r c , r d ) and (g ′ (z), p ′ (z, t), r ′ c , r ′ d ) describe the same element of M G C / M G iff there exists an element (h(z), q(z, t), s c , s d ) ∈ M G R such that (g(z), p(z, t), r c , r d ) = (g ′ (z) + h(z), p ′ (z, t) + q(z, t), r ′ c + s c , r ′ d + s d ) . Hence the equivalence classes for g(z) are elements of M G C /M G R . The extension of M G R lies in S 1 . Thus r c and r d are defined up to an element in S 1 . So we obtain a description of M G C / M G R as a (R + ) 2 -bundle over M G C /M G R .
As M G C /M G R is diffeomorphic to a vector space (and hence simply connected), this bundle is trivial. The diffeomorphism can be described by mapping (g(z), p(z, t), r c , r d ) onto (g(z), r c , r d ), hence by forgetting the path p(z, t).
Now we investigate the quotients M G/ Fix(σ).
The group Fix(σ) consists of elements (g, p, r c , r d ) ∈ M G such that {g, p ∈ Fix(σ), r c ∈ ±1, r d ∈ ±1}. As {r c , r d } ∈ ±1, this is a covering with four leaves. The details of the argument follow the description found in [Pop05] for the smooth C ∞ -setting. Let H ⊂ M G C be a real form of non-compact type. The description of the space H/Fix(σ) ⊂ M G C /Fix(σ) follows similarly. This yields a proof of the following theorem: Theorem 7.7. The space H/Fix(σ) is diffeomorphic to a vector space.
In section 4.3 we introduced the pointwise exponential function Mexp : M g −→ M G. This exponential function can be extended to an exponential function on the Kac-Moody algebra M g.
Mexp : M g −→ M G Via Mexp the central extension Cc corresponds to the fiber of the C * -bundle and the Cd-term corresponds to the C * -factor of the semidirect product.
For the group M G we can describe the exponential function as follows: For the straightforward proof one has to check that this defines a 1-parameter subgroup whose differential at the identity is f + r c c.
Here ω(u) denotes the shift of the argument by ω.
• c generates the center, thus Ad(x)c := c.
• Ad(x)u follows by integrating the Ad-action.
• To calculate Ad(g)(d), we use the Ad-invariance of the Lie bracket. As [d, v] = v ′ we get for all v ∈ L alg g σ : with h ∈ L alg g σ and {µ, ν} ∈ R.
Thus we are left with the calculation of µ. To this end we use the property that Ad acts by isometries. This results in µ = 1 2 g ′ g −1 , g ′ g −1 . More details can be found in [PS86,HPTT95,Pop05,Pop06].
Theorem 7.10 (Polarity of the Adjoint action). Let H l,r ⊂ Xg σ , {l, r} ∈ R\{0} denote the intersection of the sphere with radius −l 2 with the horosphere r d = r. The restriction of the Adjoint action to H l,r is polar.
Proof. The restriction of the Adjoint action to H l,r coincides with the gauge action on Xg. Hence, theorem 7.10 is a direct consequence of theorem 5.3.
C.-L. Terng describes how to associate an affine Weyl group to this gauge action [Ter95]. This is exactly the affine Weyl group of the Kac-Moody group M G. This theorem gives a complete description of the Adjoint action iff r d = 0. Surprisingly in the remaining case r d = 0 the situation is different: now the Adjoint action is reduced to the equations: Ad(x)u := gw(u)g −1 + gw(u)g −1 , g ′ g −1 c Ad(x)c := c Calculate the orbit of the constant function u ≡ 0. u is fixed by the Adjoint action as Ad(x)u := g0g −1 + g0g −1 , g ′ g −1 c = 0. Hence iff we can describe the restriction of the Adjoint action to Xg σ as some kind of polar action then the associated Weyl group has to be necessarily of spherical type. Furthermore the action is clearly not proper Fredholm. Hence the Hilbert-space version is not covered by Terng's results. We use the regularity independent notation: We define a flat of finite type to be a flat t ⊂ L(g, σ) such that t is the restriction of a flat in L(g, σ). Hence all flats of finite type are conjugate in L(G, σ) and as the orbits of L(G, σ) and L(G, σ) coincide on L(g, σ) also in L(G, σ). Hence any flat of finite type in L(g, σ) is isomorphic to t 0 ⊂ g, where we choose g to denote the subalgebra of constant loops. Using the usual notion for regular and singular elements we find that the associated Weyl group is the spherical Weyl group of g.
Remark 7.11. From a geometric point of view this different behavior is related to the fact that the hyperplane defined by r d = 0 corresponds to the spherical building at infinity while the space r d = 0 corresponds to the spherical building at infinity. For further details confer [Fre09], [Fre12c], [Fre11b].