Loop groups and string topology Lectures for the summer school algebraic groups

Göttingen, July 2005

Thomas Schick^{∗} Uni Göttingen Germany

Last compiled July 1, 2005; last edited June 30, 2005 or later

1 Introduction

Let G be a compact Lie group. Then the space LG of all maps from the circle S^{1} to G becomes a group by pointwise multiplication. Actually, there are different variants of LG, depending on the classes of maps one considers, and the topology to be put on the mapping space. In these lectures, we will always look at the space of smooth (i.e. C^{∞}) maps, with the topology of uniform convergence of all derivatives.

These groups certainly are not algebraic groups in the usual sense of the word. Nevertheless, they share many properties of algebraic groups (concerning e.g. their representation theory). There are actually analogous objects which are very algebraic (compare e.g. [1]), and it turns out that those have properties remarkably close to those of the smooth loop groups.

The lectures are organized as follows.

(1) Lecture 1: Review of compact Lie groups and their representations, basics of loop groups of compact Lie groups.

(2) Lecture 2: Finer properties of loop groups

(3) Lecture 3: the representations of loop groups (of positive energy)

The lectures and these notes are mainly based on the excellent monograph “Loop groups” by Pressley and Segal [3].

2 Basics about compact Lie groups

2.1 Definition. A Lie group G is a smooth manifold G with a group structure, such that the map G × G → G; (g, h) ↦ gh^{−1} is smooth.

∗e-mail: schick@uni-math.gwdg.de www: http://www.uni-math.gwdg.de/schick Fax: ++49 -551/39 2985

The group acts on itself by left multiplication: l_{g}(h) = gh. A vector field X ∈ Γ(T G) is called left invariant, if (l_{g})_{∗}X = X for each g ∈ G. The space of all left invariant vector fields is called the Lie algebra Lie(G). If we consider vector fields as derivations, then the commutator of two left invariant vector fields again is a left invariant vector field. This defines the Lie bracket [·, ·]: Lie(G) × Lie(G) → Lie(G); [X, Y] = XY − Y X.

By left invariance, each left invariant vector field is determined uniquely by its value at 1 ∈ G, therefore we get the identification T_{1}G ≅ Lie(G); we will frequently use both variants.

To each left invariant vector field X we associate its flow Ψ_{X}: G × ℝ → G (a priori, it might only be defined on an open subset of G × {0}). We define the exponential map

exp: Lie(G) → G; X ↦ Ψ_{X}(1, 1).

This is defined on an open subset of 0 ∈ G. The differential d_{0} exp: Lie(G) → T_{1}(G) = Lie(G) is the identity, therefore on a suitably small open neighborhood of 0, exp is a diffeomorphism onto its image.

A maximal torus T of the compact Lie group G is a Lie subgroup T ⊂ G which is isomorphic to a torus T^{n} (i.e. a product of circles) and which has maximal rank among all such. It’s a theorem that for a connected compact Lie group G and a given maximal torus T ⊂ G, an arbitrary connected abelian Lie subgroup A ⊂ G is conjugate to a subgroup of T.

2.2 Example. The group U(n) := {A ∈ M(n, ℂ) | AA^{∗} = 1} is a Lie group, a Lie submanifold of the group Gl(n, ℂ) of all invertible matrices.

In this case, T_{1}U(n) = {A ∈ M(n, ℂ) | A + A^{∗} = 0}. The commutator of T_{1}U(n) = L(U(n)) is the usual commutator of matrices: [A, B] = AB − BA for A, B ∈ T_{1}U(n).

The exponential map for the Lie group U(n) is the usual exponential map of matrices, given by the power series:

exp: T_{1}U(n) → U(n); A ↦ exp(A) = ∑_{k=0}^{∞} A^{k}/k!.

The functional equation shows that the image indeed belongs to U(n). Similarly, Lie(SU(n)) = {A ∈ M(n, ℂ) | A + A^{∗} = 0, tr(A) = 0}.

2.3 Theorem. If G is a connected compact Lie group, then exp: Lie(G) → G is surjective.

2.4 Definition. Every compact connected Lie group G can be realized as a Lie-subgroup of SU(n) for big enough n. It follows that its Lie algebra Lie(G) is a sub-Lie algebra of Lie(U(n)) = {A ∈ M(n, ℂ) | A^{∗} = −A}. Therefore, the complexification Lie(G)⊗_{ℝ} ℂ is a sub Lie algebra of Lie(U(n))⊗_{ℝ} ℂ = {A+iB | A, B ∈ M(n, ℂ), A^{∗} = −A, (iB)^{∗} = −(iB)} = M(n, ℂ). Note that the bracket (given by the commutator) is complex linear on these complex vector spaces.

The corresponding sub Lie group G_{ℂ} of Gl(n, ℂ) (the simply connected Lie group with Lie algebra M(n, ℂ)) with Lie algebra Lie(G) ⊗_{ℝ} ℂ is called the complexification of G.

G_{ℂ} is a complex Lie group, i.e. the manifold G_{ℂ} has a natural structure of a complex manifold (charts with holomorphic transition maps), and the composition G_{ℂ} × G_{ℂ} → G_{ℂ} is holomorphic.

2.5 Exercise. Check the assertions made in Definition 2.4, in particular about the complex structure.

2.6 Example. The complexified Lie algebra Lie(SU(n))⊗_{ℝ}ℂ = {A ∈ M(n, ℂ) | tr(A) = 0}, and the complexification of SU(n) is Sl(n, ℂ).

2.7 Definition. Let G be a compact Lie group. It acts on itself by conjugation: G × G → G; (g, h) ↦ ghg^{−1}.

For fixed g ∈ G, we can take the differential of the corresponding map h ↦ ghg^{−1} at h = 1. This defines the adjoint representation ad: G → Gl(Lie(G)).

We now decompose Lie(G) into irreducible sub-representations for this action, Lie(G) = 𝔤_{1} ⊕ ⋯ ⊕ 𝔤_{k}. Each of these are Lie subalgebras, and we have [𝔤_{i}, 𝔤_{j}] = 0 if i ≠ j.

G is called semi-simple if non of the summands is one-dimensional, and simple if there is only one summand, which additionally is required not to be one-dimensional.

2.8 Remark. The simply connected simple compact Lie groups have been classified, they consist of SU(n), SO(n), the symplectic groups Sp_{n} and five exceptional groups (called G_{2}, F_{4}, E_{6}, E_{7}, E_{8}).

2.9 Definition. Let G be a compact Lie group with a maximal torus T. G acts (induced from conjugation) on Lie(G) via the adjoint representation, which induces a representation on Lie(G)⊗_{ℝ}ℂ =: 𝔤_{ℂ}. This Lie algebra contains the Lie algebra 𝔱_{ℂ} of the maximal torus, on which T acts trivially (since T is abelian).

Since T is a maximal torus, it acts non-trivial on every non-zero vector of the complement.

As every finite dimensional representation of a torus, the complement decomposes into a direct sum ⨁ 𝔤_{α}, where on each 𝔤_{α}, t ∈ T acts via multiplication with α(t) ∈ S^{1}, where α: T → S^{1} is a homomorphism, called the weight of the summand 𝔤_{α}.

We can translate the homomorphisms α: T → S^{1} into their derivative at the identity, thus getting a linear map α′: 𝔱 → ℝ, i.e. an element of the dual space 𝔱^{∗}, they are related by α(exp(x)) = e^{iα′(x)}.

This way, we think of the group of characters T̂ = Hom(T, S^{1}) as a lattice in t^{∗}, called the lattice of weights. It contains the set of roots, i.e. the non-zero weights occurring in the adjoint representation of G.

2.10 Remark. As 𝔤_{ℂ} is the complexificatoin of a real representation, if α is a root of G, so is −α, with 𝔤_{−α} = \overline{𝔤_{α}}.

It is a theorem that the subspaces 𝔤_{α} are always 1-dimensional.

2.11 Example. For U(n), Lie(U(n))_{ℂ} = M(n, ℂ). A maximal torus is given by the diagonal matrices (with diagonal entries in S^{1}). We can then index the roots by pairs (i, j) with 1 ≤ i ≠ j ≤ n. We have α_{ij}(diag(z_{1}, …, z_{n})) = z_{i}z_{j}^{−1} ∈ S^{1}. The corresponding subspace 𝔤_{ij} consists of matrices which are zero except for the (i, j)-entry. As an element of 𝔱^{∗} it is given by the linear map ℝ^{n} → ℝ; (x_{1}, …, x_{n}) ↦ x_{i} − x_{j}.

2.12 Definition. According to Remark 2.10, each of the spaces 𝔤_{α} is 1-dimensional. Choose a vector 0 ≠ e_{α} ∈ 𝔤_{α}, such that e_{−α} = \overline{e_{α}}. Then h_{α} := −i[e_{α}, e_{−α}] ∈ 𝔱 is non-zero. We can normalize the vector e_{α} in such a way that [h_{α}, e_{α}] = 2ie_{α}. Then h_{α} is canonically determined by α, it is called the coroot associated to α.

We call G a simply knitted Lie group, if there is a G-invariant inner product on Lie(G) such that ⟨h_{α}, h_{α}⟩ = 2 for all roots α.

This inner product gives rise to an isomorphism 𝔱 ≅ 𝔱^{∗}; this isomorphism maps h_{α} to α.

2.13 Example. For SU(2), T = {diag(z, z^{−1}) | z ∈ S^{1}}, there is only one pair of roots α, −α, obtained by restriction of the roots α_{12} and α_{21} of U(2) to the (smaller) maximal torus of SU(2). We get α(diag(z, z^{−1})) = z^{2}, e_{α} = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ),

h_{α} = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ).

All the groups U(n) and SU(n) are simply knitted, as well as SO(2n). The canonical inner product of Lie(U(n))_{ℂ} = M(n, ℂ) is given by ⟨A, B⟩ = −tr(A^{∗}B)/2.

2.14 Proposition. Given any semi-simple Lie group, the Lie algebra is spanned by the vectors e_{α}, e_{−α}, h_{α} as α varies over the set of (positive) roots.

For each such triple one can define a Lie algebra homomorphism Lie(SU(2)) → Lie(G) which maps the standard generators of Lie(SU(2)) to the chosen generators of Lie(G).

We get the technical useful result that for a semi-simple compact Lie group G the Lie algebra Lie(G) is generated by the images of finitely many Lie algebra homomorphisms from Lie(SU(2)) to L(G).

2.15 Definition. Let G be a compact Lie group with maximal torus T. Let N(T) := {g ∈ G | gT g^{−1} ⊂ T } be the normalizer of T in G. The Weyl group W (T) := N(T)/T acts on T by conjugation, and via the adjoint representation also on 𝔱 and then on 𝔱^{∗}. It is a finite group.

This action preserves the lattice of weights T̂ ⊂ 𝔱^{∗}, and also the set of roots. Given any root α of G, the Weyl group contains an element s_{α} of order 2. It

acts on 𝔱 by reflection on the hyperplane H_{α} := {X | α(X) = 0}. More precisely we have s_{α}(X) = X − α(X)h_{α}, where h_{α} ∈ 𝔱 is the coroot associated to the root α.

The reflections s_{α} together generate the Weyl group W (G).

The Lie algebra 𝔱 is decomposed into the union of the root hyperplanes H_{α} and into their complement, called the set of regular elements. The complement decomposes into finitely many connected components, which are called the Weyl chambers. One chooses one of these and calls it the positive Weyl chamber. The Weyl group acts freely and transitively on the Weyl chambers.

A root of G is called positive or negative, if it assumes positive or negative values on the positive chamber. A positive root α is called simple, if its hyperplane H_{α} is a wall of the positive chamber.

2.16 Example. For the group U(n), the Weyl group is the symmetric group S_{n}, it acts by permutation of the diagonal entries on the maximal torus. Lifts of the elements of W (U(n)) to N(T) ⊂ U(n) are given by the permutation matrices.

For SU(n), the Weyl group is the alternating group S_{n}, again acting by permutation of the diagonal entries of the maximal torus.

Since these groups are simply knitted, we can depict their roots, coroots etc. in a simply Euclidean picture of 𝔱 ≅ 𝔱^{∗}. We look at the case G = SU(3) where Lie(T) is 2-dimensional.

In the lecture, a picture is drawn.

In general, the positive roots of U(n) or SU(n) can be chosen to be the roots α_{ij} with i < j.

2.17 Theorem. Let G be a compact semi-simple Lie group. There is a oneto-one correspondence between the irreducible representations of G and the set of dominant weights, where a weight α is called dominant if α(h_{β}) ≥ 0 for each positive root β of G.

Proof. We don’t proof this theorem here; we just point out that the representation associated to a dominant weight is given in as the space of holomorphic sections of a line bundle over a homogeneous space of G_{ℂ} associated to the weight.

2.18 Remark. One of the goals of these lectures is to explain how this results extends to loop groups.

3 Basics about loop groups

3.1 Definition. An infinite dimensional smooth manifold (modeled on a locally convex complete topologically vector space X) is a topological space M together with a collection of charts x_{i}: U_{i} → V_{i}, with open subset U_{i} of M and V_{i} of X and a homeomorphism x_{i}, such that the change of coordinate maps x_{j}∘x_{i}^{−1} : V_{i} → V_{j} are smooth maps between the topological vector space X.

Differentiability of a map f: X → X is defined in terms of convergence of difference quotients, if it exists, the differential is then a map

Df: X × X → X; (v, w) ↦ lim_{t→0} \frac{f(u + tv) − f(u)}{t}

(where the second variable encodes the direction of differentiation), iterating this, we define higher derivatives and the concept of smooth, i.e. C^{∞} maps.

3.2 Definition. Let G be a compact Lie group with Lie algebra Lie(G) = T_{1}G e( a finite dimensional vector space). For us, the model topological vector space X will be X = C^{∞}(S^{1}, Lie(G)). Its topology is defined by the collection of semi-norms q_{i}(f) := sup_{x∈S^{1}}{|∂^{i}ϕ/∂t^{k}(x)|} for any norm on Lie(G).

This way, X is a complete separable (i.e. with a countable dense subset) metrizable topological vector space. A sequence of smooth functions ϕ_{k}: S^{1} → Lie(G) converges to ϕ: S^{1} → Lie(G) if and only if the functions and all their derivatives converge uniformly.

3.3 Lemma. There is a canonical structure of a smooth infinite dimensional manifold on LG. A chart around the constant loop 1 is given by the exponential map C^{∞}(S^{1}, U) → LG; χ ↦ exp ∘χ where U is a sufficiently small open neighborhood of 0 ∈ T_{1}(G). Charts around any other point f ∈ LG are obtained by translation with f, using the group structure on LG.

We define the topology on LG to be the finest topology (as many open sets as possible) such that all the above maps are continuous.

It is not hard to see that the transition functions are then actually smooth, and that the group operations (defined pointwise) are smooth maps.

3.4 Exercise. Work out the details of Lemma 3.3, and show that it extends to the case where S^{1} is replaced by any compact smooth manifold N.

3.5 Remark. There are many variants of the manifold LG of loops in G. Quite useful are versions which are based on maps of a given Sobolev degree, one of the advantages being that the manifolds are then locally Hilbert spaces.

3.6 Exercise. Show that for G = SU(2) the group LG is connected, but the exponential map is not surjective

3.7 Exercise. There are other interesting infinite dimensional Lie groups. One which is of some interest for loop groups is Diffeo(S^{1}), the group of all diffeomorphisms of S^{1} (and also its identity component of orientation preserving diffeomorphism).

Show that this is indeed a Lie group, with Lie algebra (and local model for the smooth structure) V ect(S^{1}), the space of all smooth vector fields. exp: V ect(S^{1}) → Diffeo(S^{1}) maps a vector field to the (time 1) flow generated by it.

Show that there are no neighborhoods U ⊂ V ect(S^{1}) of 0 and V ⊂ Diffeo(S^{1}) of id_{S^{1}} such that exp |: U → V is injective or surjective.

3.8 Lemma. Consider the subgroup of based loops Ω(G) = {f: S^{1} → G ∈ LG | f(1) = 1} ⊂ LG, and the subgroup of constant loops G ⊂ LG.

The multiplication map G × Ω(G) → LG is a diffeomorphism.

Proof. The inverse map is given by LG → G × Ω(G); f ↦ (f(1), f(1)^{−1}f). Clearly the two maps are continuous and inverse to each other. The differentiable structure of Ω(G) makes the maps smooth.

3.9 Corollary. It follows that the homotopy groups of LG are easy to compute (in terms of those of G); the maps of Lemma 3.8 give an isomorphism π_{k}(LG) ≅ π_{k}(G) ⊕ π_{k−1}(G).

In particular, LG is connected if and only if G is connected and simply connected, else the connected components of G are parameterized by π_{0}(G) × π_{1}(G).

3.10 Definition. Embed the compact Lie group G into U(n). Using Fourier decomposition, we can now write the elements of LG in the form f(z) = ∑ A_{k}z^{k} with A_{k} ∈ M(n, ℂ).

We define now a number of subgroups of LG.

(1) L_{pol}G consists of those loops with only finitely many non-zero Fourier coefficients. It is the union (over N ∈ ℕ) of the subsets of functions of degree ≤ N. The latter ones are compact subsets of M(n, ℂ)^{N}, and we give L_{pol}G the direct limit topology.

(2) L_{rat}G consists of those loops which are rational functions f(z) (without poles on {|z| = 1}).

(3) L_{an}G are those loops where the series ∑ A_{k}z^{k} converges for some annulus r ≤ |z| ≤ 1/r with 0 < r < 1. For fixed 0 < r < 1 this is Banach Lie group (of holomorphic functions on the corresponding annulus), with the topology of uniform convergence. We give L_{an}G the direct limit topology.

3.11 Exercise. In general, L_{pol}G is not dense in G. Show this for the case G = S^{1}.

3.12 Proposition. If G is semi-simple, then L_{pol}G is dense in LG.

Proof. Set H := \overline{L_{pol}}G, and V := \{X ∈ C^{∞}(S^{1}, Lie(G) | exp(tX) ∈ H∀t ∈ ℝ}) ⊂ C^{∞}(S^{1}, Lie(G)). Then V is a vector space, as

exp(X + Y) = lim_{k→∞} (exp(X/k) exp(Y/k))^{k}

converges in the C^{∞}-topology if X and Y are close enough to 0. Since exp is continuous, V is a closed subspace. It remains to check that V is dense in L𝔤.

Because of Proposition 2.14 it suffices to check the statement for SU(2). Here, we first note that the elements X_{n}: z ↦ ( \begin{matrix}0 & z^{n} \\ −z^{−n} & 0 \end{matrix} ) and Y_{n}: z ↦ ( \begin{matrix}0 & iz^{n} \\ iz^{−n} & 0 \end{matrix} )

belong to V, since X_{n}^{2} = Y_{n}^{2} = −1, so that exp(tX_{n}) = ∑_{k}(tX_{n})^{k}/k! actually is a family of polynomial loops. By linearity and the fact that V is closed, every loop of the form z ↦ ( \begin{matrix}0 & f(z)+ig(z) \\ −f(z)+ig(z) & 0 \end{matrix} ) belongs to V, for arbitrary smooth

real valued functions f, g: S^{1} → ℝ (use their Fourier decomposition). Since V is finally invariant under conjugation by polynomial loops in LSU(2), it is all of 𝔰𝔲(2).

3.1 Abelian subgroups of LG

We have already used the fact that each compact Lie group G as a maximal torus T, which is unique upto conjugation.

3.13 Proposition. Let G be a connected compact Lie group with maximal torus T. Let A ⊂ LG be a (maximal) abelian subgroup. Then A(z) := {f(z) ∈ Gf ∈ A} ⊂ G is a (maximal) abelian subgroup of G.

In particular, we get a the maximal abelian subgroup A_{λ} for each smooth map λ: S^{1} → {T ⊂ G | T maximal torus} ≅ G/N, where N is the normalizer of T in G, with A_{λ} := {f ∈ LG | f(z) ∈ λ(z)}.

The conjugacy class of A_{λ} depends only on the homotopy class of λ. Since π_{1}(G/N) = W = N/T, [S^{1}, G/N] is the set of conjugacy classes of elements of the Weyl group W.

An element w ∈ W acts on T by conjugation, and the corresponding A_{λ} is isomorphic to {γ: ℝ → T | γ(t + 2π) = w^{−1}γ(t)w for all t ∈ ℝ.

Proof. It is clear that all evaluation maps of A have abelian image. If all the image sets are maximal abelian, then A is maximal abelian.

Since all maximal tori are conjugate, the action of G on the set of all maximal tori is transitive, with stabilizer (by definition) the normalizer N, so that this space is isomorphic to G/N.

Next, W = N/T acts freely on G/T, with quotient G/N. Since G/T is simply connected (a fact true for every connected Lie group), we conclude from covering theory that π_{1}(G/T) ≅ W. But then the homotopy classes of non basepoint preserving maps are bijective to the conjugacy classes of the fundamental group.

Given a (homotopy class of maps) λ: S^{1} → G/T, represented by w ∈ W = N/T (with a lift w′ ∈ N and with x ∈ Lie(G) such that exp(2πx) = w′), we can choose λ with λ(z) = exp(zx)T exp(−zx) (recall that the bijection between G/N and the set of maximal tori is given by conjugation of T).

We then get a bijection from A_{λ} to the twisted loop group of the assertion by sending f ∈ A_{λ} to f̃: ℝ → T : t ↦ exp(−tx)f(t) exp(tx).

3.14 Example. The most obvious maximal abelian subgroup of a compact Lie group G with maximal torus T is LT, which itself contains in particular T (as subgroup of constant loops).

3.15 Exercise. Find other maximal abelian subgroups of LG.

3.16 Example. If G = U(n) with maximal torus T, its Weyl group W is isomorphic to the symmetric group S_{n}. Given a cycle w ∈ W = S_{n}, the corresponding maximal abelian subgroup A_{w} ⊂ LU(n) is isomorphic to LS^{1}.

More generally, if w is a product of k cycles (possibly of length 1), then A_{w} is a product of k copies of LS^{1}.

Proof. We prove the statement if w consists of one cycle (of length n). Then, in the description of Proposition 3.13, A_{w} consists of functions ℝ → T which are periodic of period 2πl. Moreover, the different components are all determined by the first one, and differ only by a translation in the argument of 2π or a multiple of 2π.

3.17 Proposition. If G is semi-simple and compact, then LG_{0}, the component of the identity of LG, is perfect.

4 Root system and Weyl group of loop groups

4.1 Definition. Let G a compact Lie group with maximal torus T. Consider its complexified Lie algebra Lie(LG)_{ℂ} = L𝔤_{ℂ}.

It carries the action of S^{1} by reparametrization of loops: (z_{0} · X)(z) := X(z_{0}z).

We get a corresponding decomposition of L𝔤_{ℂ} = ⨁_{k∈ℤ} 𝔤_{ℂ}z^{k} into Fourier components (the sums has of course to be completed appropriately).

The action of S^{1} used in the above decomposition still commutes with the adjoint representation of the subgroup T of constant loops, so the summands can further be decomposed according to the weights of the action of T, to give a decomposition

L𝔤_{ℂ} = ⊕_{(k,α)∈ℤ×T̂}𝔤_{(k,α)}z^{k}.

The index set ℤ × T̂ is the Pontryagin dual of S^{1} × T (i.e. the set of all homomorphisms S^{1} × T → S^{1}). Those homomorphisms which occur (now also with possibly α = 0) are called the roots of LG.

The (infinite) set of roots of LG is permuted by the so called affine Weyl group

W_{aff} = N(T × S^{1})/(T × S^{1}),

considered inside the semidirect product LG ⋊ S^{1}, where we use the action of S^{1} by reparametrization (rotation) of loops on LG to construct the semidirect product. This follows because we decompose L𝔤_{ℂ} as a representation of LG⋊S^{1} with respect to the subgroup T × S^{1}..

4.2 Proposition. W_{aff} is a semidirect product Hom(S^{1}, T) ⋊ W, where W is the Weyl group of G, with its usual action on the target T.

Proof. Clearly Hom(S^{1}, T) is a subgroup of LG which conjugates the constant loops with values in T into itself, and the action of the constant loops with values in the normalizer N does the same (and factors through W). It is also evident that all of S^{1} belongs to the normalizer of T × S^{1}.

On the other hand, if R_{z_{0}} ∈ S^{1} ∈ LG ⋊ S^{1} acts by rotation by z_{0} ∈ S^{1}, then for f ∈ LG we get f^{−1}R_{z−0}f = f^{−1}(·)f(·z_{0})R_{z_{0}}.

This belongs to T ×S^{1} if and only if z ↦ f(z)^{−1}f(zz_{0}) is a constant element of T for each z_{0}, i.e. if and only if z ↦ f(1)^{−1}f(z) is a group homomorphism z → T. Additionally, f conjugates T to itself if and only if f(1) ∈ N. Therefore, N(T × S^{1}) is the product of N, Hom(S^{1}, T) and S^{1} inside LG ⋊ S^{1}, and the quotient by T × S^{1} is as claimed.

4.3 Definition. We think of the weights and roots of LG not as linear forms on ℝ×𝔱 (derivatives of elements in Hom(S^{1} ×T, S^{1})), but rather as affine linear functions on 𝔱, where we identify 𝔱 with the hyperplane 1 × 𝔱 in ℝ × 𝔱; this explains the notation “affine roots”.

Moreover, the group W_{aff} acts linearly on ℝ × 𝔱 and preserves 1 × 𝔱, where λ ∈ Hom(S^{1}, T) acts by translation by λ′(1) ∈ 𝔱.

An affine root (k, α) is (for α ≠ 0) determined (upto sign) by the affine hyperplane H_{k,α} := {x ∈ 𝔱 | α(x) = −k} ⊂ 𝔱 which is the set where it vanishes.

The collection of these hyperplanes is called the diagram of LG. It contains as the subset consisting of the H_{0,α} the diagram of G

Recall that the connected components of the complement of all H_{0,α} were called the chambers of G, and we choose one which we call the positive chamber. The components of the complement of the diagram of LG are called alcoves. Each chamber contains a unique alcove which touches the origin, and this way the positive chamber defines a positive alcove, the set {x ∈ ⊔ | 0 < α(x) < 1 for all positive roots α}. An affine root is called positive or negative, if it has positive or negative values at the positive alcove. The positive affine roots corresponding to the walls of the positive alcove are called the simple affine roots.

4.4 Example. The diagram for SU(3) is the tessellation of the plane by equilateral triangles.

In general, if G is a simple group, then each chamber is a simplicial cone, bounded by the l hyperplanes H_{0,α_{1}}, …, H_{0,α_{l}} (where α_{1}, …, α_{l} are the simple roots of G). There is a highest root α_{l+1} of G, which dominates all other roots (on the positive chamber). The positive alcove is then an l-dimensional simplex cut out of the positive chamber by the wall H_{1,−α_{l+1}}, and we get l + 1 simple affine roots of LG, (0, α_{1}), …, (0, α_{l}), (1, −α_{l+1}).

In general, if G is semi-simple with q simple factors, the positive alcove is a product of q simplices, bounded by the walls of the simple roots of G, and walls H_{1,−α_{i}}, i = l + 1, …, l + q being the highest weights of the irreducible summands of the adjoint representation.

4.5 Proposition. Let G be a connected and simply connected compact Lie group. Then W_{aff} is generated by reflections in the hyperplanes (the reflections in the walls of the positive alcove suffice), and it acts freely and transitively on the set of alcoves.

Proof. From the theory of compact Lie groups, we know that W is generated by the reflections at the H_{0,α}, and that Hom(S^{1}, 𝔱) is generated by the coroots h_{α}.

Recall that the reflection s_{α} was given by s_{α}(x) = x − α(x)h_{α}. Recall that we normalized such that α(h_{α}) = 2, therefore −kh_{α}/2 ∈ H_{k,α}.

Now the reflection s_{k,α} in the hyperplane H_{k,α} is given by s_{k,α}(x) = x + kh_{α}/2 − α(x + kh_{α}/2)h_{α} − kh_{α}/2 = s_{α}(x) − kh_{α}. Since s_{α} ∈ W and h_{α} in Hom(S^{1}, T) (identified with the value of its derivative at 1), s_{k,α} ∈ W_{aff} = Hom(S^{1}, T) ⋊ W.

To show that these reflections generate W_{aff}, it suffices to show that they generate the translation by h_{α}. But this is given by s_{a,−α}s_{0,α}.

We now show that W_{aff} acts transitively on the set of alcoves. For an arbitrary alcove A, we have to find γ ∈ W_{aff} such that γA is the positive alcove C_{0}. Now the orbit W_{aff}a of a point p ∈ A is a subset S of 𝔱 without accumulation points. Choose a point c ∈ C_{0} and one of the points b ∈ S with minimal distance to c. If b would not belong to C_{0}, then b and c are separated by a wall of C_{0}, in which we can reflect b to obtain another point of S, necessarily closer to c than b.

Since W acts freely on the set of chambers, W_{aff} acts freely on the set of alcoves (since each element of W preserves the distance to the origin, and each translation moves the positive alcove away from the origin, only elements of W could stabilize the positive alcove).

5 Central extensions of LG

We want to study the representations of LG for a compact Lie group G. It turns out, however, that most of the relevant representations are no honest representations but only projective representations., i.e. U_{f}U_{g} = c(f, g)U_{fg} for every f, g ∈ LG, where U_{f} is the operator by which f acts, and c(f, g) ∈ S^{1} is a scalar valued function (a cocycle).

More precisely, the actions we consider are actions of central extensions of LG (in some sense defined by this cocycle). We don’t want to go into the details of the construction and classification of these central extensions, but only state the main results.

Let G be a compact connected Lie group.

(1) LG has many central extensions 1 → S^{1} → L̃G → LG → 1.

(2) The corresponding Lie algebras L̃𝔤 are classified as follows: for every symmetric invariant bilinear form ⟨·, ·⟩: 𝔤 × 𝔤 → ℝ we get a form

ω: L𝔤 × L𝔤 → ℝ; (X, Y) ↦ \frac{1}{2π} ∫_{0}^{2π}⟨X(z), Y′(z)⟩ dz.

Then L̃𝔤 = ℝ ⊕ L𝔤 with bracket

[(a, X), (b, Y)] = (ω(X, Y), [X, Y]).

(3) The extension 0 → ℝ → L̃𝔤 → L𝔤 → 0 with bracket given by the inner product ⟨, ⟩ on 𝔤 corresponds to an extension of Lie groups 1 → S^{1} → L̃G → LG → 1 if and only if ⟨h_{α}, h_{α}⟩ ∈ 2ℤ for every coroot h_{α} of G.

(4) If this integrality condition is satisfied, the extension L̃G is uniquely determined. Moreover, there is a unique action of Diffeo^{+}(S^{1}) on L̃G which covers the action on LG. In particular, there is a induced canonical lift of the action of S^{1} on LG by rotation of the argument to L̃G.

(5) The integrality condition is satisfied if and only if ω/2π, considered as an invariant differential form on LG (which is closed by the invariance of ⟨, ⟩ and therefore of ω), lifts to an integral cohomology class. It then represents the first Chern class of the principle S^{1}-bundle S^{1} → L̃G → LG over LG. It follows that the topological structure completely determines the group extension.

(6) If G is simple and simply connected there is a universal central extension 1 → S^{1} → L̃G → LG → 1 (universal means that there is a unique map of extensions to any other central extension of LG).

If G is simple, all invariant bilinear forms on 𝔤 are proportional, and the universal extension corresponds to the smallest non-trivial one which satisfies the integrality condition. We call this also the basic inner product and the basic central extension.

(7) For SU(n) and the other simply laced groups, the basic inner product is the canonical inner product such that ⟨h_{α}, h_{α}⟩ = 2 for every coroot α.

(8) There is a precise formula for the adjoint and coadjoint action of suitable elements, compare Lemma 6.10.

6 Representations of loop groups

6.1 Definition. A representation of a loop group LG (or more generally any topological group) is for us given by a locally convex topological vector space V (over ℂ) with an action

G × V → V; (g, v) → gv

which is continuous and linear in the second variable.

Two representations V_{1}, V_{2} are called essentially equivalent, if they contain dense G-invariant subspaces V′_{1} ⊂ V_{1}, V′_{2} ⊂ V_{2} with a continuous G-equivariant bijection V′_{1} → V′_{2}

The representation V is called smooth, if there is a dense subspace of vectors v ∈ V such that the map G → V; g ↦ gv is smooth —such vectors are called smooth vectors of the representation.

A representation V is called irreducible if it has no closed invariant subspaces.

6.2 Example. The actions of S^{1} on C^{∞}(S^{1}) ,C^{(}S^{1}) and L^{2}(S^{1}) by rotation of the argument are all equivalent and smooth.

However, rotation of the argument does not define an action in our sense of S^{1} on L^{∞}(S^{1}) because the corresponding map S^{1} × L^{∞}(S^{1}) → L^{∞}(S^{1}) is not continuous.

6.3 Remark. Given a representation V of LG and z_{0} ∈ S^{1} (or more generally any diffeomorphism of S^{1}, z_{0} gives rise to a diffeomorphism by translation), then we define a new representation ϕ^{∗}V by composition with the induced automorphism of LG.

We are most interested in representations which are symmetric, meaning that ϕ^{∗}V ≅ V. Actually, we require a somewhat stronger condition in the following Definition 6.4.

6.4 Definition. When we consider representation V of LG, we really want to consider actions of LG⋊S^{1}, i.e. we want an action of S^{1} on V which intertwines the action of LG; for z_{0} ∈ S^{1} we want operators R_{z_{0}} on V such that R_{z_{0}}fR_{z_{0}}^{−1}v = f(· + z_{0})v for all v ∈ V, f ∈ LG.

Moreover, we will study projective representations, i.e. representations such that f_{1} · (f_{2} · v) = c(f_{1}, f_{2})(f_{1}f_{2}) · v with c(f_{1}, f_{2}) ∈ ℂ \ {0}.

More precisely, these are actions of a central extension 1 → ℂ^{×} → L̃G → LG → 1.

Since S^{1} acts on V, we get a decomposition V = \overline{⨁_{k∈ℤ} V (k)}, where z ∈ S^{1} acts on V (k) by multiplication with z^{−k}.

We say V is a representation of positive energy, if V (k) = 0 for k < 0.

6.5 Example. The adjoint representation of LG on L𝔤, or the canonical representation of LSU(n) on the Hilbert space L^{2}(S^{1}, ℂ^{n}) are not of positive energy, nor of negative energy. However, they are not irreducible.

6.6 Remark. One can always modify the action of S^{1} on a representation V of LG by multiplication with a character of S^{1}, so that “positive energy” and “energy bounded below” are more or less the same.

The complex conjugate of a representation of negative energy is a representation of positive energy.

6.7 Proposition. An irreducible unitary representation V of L̃G ⋊ S^{1} which is of positive energy (and this makes only sense with the action of the extra circle) is also irreducible as a representation of L̃G.

Proof. Let T be the projection onto a proper L̃G-invariant summand of V. This operator commutes with the action of L̃G. Let R_{z} be the operator through which z ∈ S^{1} acts on V. We define the bounded operators T_{q} := ∫_{S^{1}} z^{q}R_{z}T R_{z}^{−1} dz. They all commute with L̃G, and T_{q} maps V (k) to V (k + q).

Let m be the lowest energy of V. Then T_{q}(V (m)) = 0 for all q < 0. Since V is irreducible, V is generated as a representation of LG ⋊ S^{1} by V (m). Since V (m) is S^{1}-invariant, T_{q}(V) = 0 for q < 0. Since T_{−q} = T_{q}^{∗}, we even have T_{q} = 0 for all q ≠ 0. Now, the T_{q} are the Fourier coefficients of the loop z ↦ R_{z}T R_{z}^{−1}. It follows that this loop is constant, i.e. that T commutes also with the action of S^{1}. But since V was irreducible, this implies that T = 0.

The representation of positive energy of a loop group behave very much like the representation of a compact Lie group. This is reflected in the following theorem.

6.8 Theorem. Let G be a compact Lie group. Let V be a smooth representation of positive energy of L̃G ⋊ S^{1}. Then, upto essential equivalence:

(1) If V is non-trivial then it does not factor through an honest representation of LG, i.e. is truly projective.

(2) V is a discrete direct sum of irreducible representations (of positive energy)

(3) V is unitary

(4) The representation extends to a representation of L̃G⋊Diffeo^{+}(S^{1}), where Diffeo^{+}(S^{1}) denotes the orientation preserving diffeomorphisms and contains S^{1} (acting by translation).

(5) V extends to a holomorphic projective representation of LG_{ℂ}.

Granted this theorem, it is of particular importance to classify the irreducible representations of positive energy

6.9 Definition. Let T_{0}×T ×S^{1} ⊂ L̃G⋊S^{1} be a “maximal torus”, with T_{0} = S^{1} the kernel of the central extension, T a maximal torus of G and S^{1} the rotation group. We can then refine the energy decomposition of any representation V to a decomposition

V = \overline{⨁_{n,α,h∈ℤ×T̂×ℤ} V_{n,α,h}}

according to the characters of T_{0} × T × S^{1}. The characters which occur are called the weights of V. n is called the energy and h the level.

6.10 Lemma. The action of ξ ∈ Hom(S^{1}, T) on a weight (n, α, h) is given by

ξ(n, α, h) = (n + α(ξ) + h |ξ|^{2} /2, α + hξ^{∗}, h)

where we identify ξ with ξ′(1) ∈ 𝔱.

Moreover, the norm is obtained from the inner product on L𝔤 which corresponds to the central extension L̃G, and ξ^{∗} ∈ T̂ is the image of ξ under the map 𝔱 → 𝔱^{∗} defined by this inner product.

6.11 Remark. Note that T_{0} commutes with every element of L̃G ⋊ S^{1}. Consequently, each representation decomposes into subrepresentations with fixed level, and an irreducible representation has only one level h.

The level is a measure for the “projectivity” of the representation; it factors through an honest representation of LG if and only if the level is 0.

The weights of a representation are permuted by the normalizer of T_{0}×T ×S^{1}, hence by the affine Weyl group W_{aff} = Hom(S^{1}, T) ⋊ W (where W is the Weyl group of G).

6.12 Definition. Given a root (n, α) of LG, we define the coroot (−n |h_{α}|^{2} /2, h_{α}) ∈ ℝ⊕𝔱 ⊂ ℝ⊕𝔱⊕ℝ, where we use again the inner product for the central extension L̃G.

Our irreducible representation have a number of additional important properties.

6.13 Theorem. Let V be a smooth irreducible representation of LG of positive energy (i.e. we really take a representation of L̃G ⋊ S^{1}).

(1) Then V is of finite type, i.e. for each energy n the subspace V (n) is finite dimensional. In particular, each weight space E_{h,λ,n} is finite dimensional.

(2) V has a unique lowest weight (h, λ, n) with E_{h,λ,n} ≠ 0. Lowest weight means by definition, that for each positive root (α, m) the character (h, λ− α, n − m) does not occur as a weight in V.

This lowest weight is antidominant, i.e. for each positive coroot (−m |h_{α}|^{2} /2, h_{α}) we have ⟨(h, λ, n), (−m |h_{α}|^{2} /2), h_{α}, −0⟩ = λ(h_{α}) − hm |h_{α}|^{2} /2 ≤ 0.

Since we have in particular to consider the positive roots (α, 0) and (−α, 1)

(for each positive root α of G), this is equivalent to

−h |h_{α}|^{2} /2 ≤ λ(h_{α}) ≤ 0  (6.14)

for each positive root α of G.

(3) There is a bijection between isomorphism class of irreducible representations of L̃G ⋊ S^{1} as above and antidominant weights.

6.15 Corollary. If the level h = 0, only λ = 0 satisfies Inequality (6.14). In other words, among the representations considered here, only the trivial representation is an honest representation of LG, all others are projective.

For a given level h, there are only finite possible antidominant weights (with n = 0), because the h_{α} generate 𝔱.

6.16 Example. If G is simple and we look at an antidominant weight (h, λ, 0), then −λ is a dominant weight in the usual sense of G, i.e. contained in the corresponding simplicial cone in 𝔱^{∗}, but with the extra condition that it is contained in the simplex cut off by {μ | μ(α_{0}) = h}, where α_{0} is the highest weight of G.

We get in particular the so called fundamental weights

(1) (1, 0, 0)

(2) (⟨ω_{i}, α_{0}⟩, −ω_{i}, 0), with ω_{i} the fundamental weights of G determined by ω_{i}(h_{α_{j}}) = δ_{ij}.

The antidominant weights are exactly the linear combinations of the fundamental weights with coefficients in ℕ ∪ {0}.

Given an irreducible representation V of L̃G ⋊ S^{1} of lowest weight (h, λ, 0), to get a better understanding of the we want to determine which other weights occur in V (or rather, we want to find restrictions for those weights).

First observation: the whole orbit under W_{aff} occurs. This produces, for the η ∈ Hom(S^{1}, T), the weights (h, λ + hη^{∗}, λ(η) + h |η|^{2} /2),

6.17 Example. If G = SU(2), we have the isomorphism (T̂ ⊂ 𝔱) ≅ (ℤ ⊂ ℝ), where ( \begin{matrix}2πit & 0 \\ 0 & −2πit \end{matrix} ) ∈ 𝔱 is mapped to t ∈ ℝ, 𝔱^{∗} is identified with 𝔱 using the standard inner product and this way the character diag(z, z^{−1}) ↦ z^{μ} in T̂ is mapped to μ ∈ ℤ.

Under this identification, the lowest weight α_{0} is identified with 1 ∈ ℤ.

There are exactly two fundamental weights. The W_{aff}-orbit of the weight (1, 0, 0) is (with this identification of T̂ with ℤ) {(1, 2k, m) | (2k)^{2} = 2m}, the set of all weights of the corresponding irreducible representations is {(1, 2k, m) | (2k)^{2} ≤ 2m}. Similarly, the orbit of (1, −1, 0) is {(1, 2k + 1, m) | (2k + 1)^{2} = 2m + 1}, the set of all weights is {(1, 2k + 1, m) | (2k + 1)^{2} ≤ 2m + 1}.

In general (for arbitrary G), the orbit of the lowest weight (h, μ, m) (with |(h, μ, m)|^{2} = |μ|^{2} + 2mh) is contained in the parabola {(h, μ′, m′) | |μ′|^{2} = |(h, μ, m)|^{2} + 2m′h}.

All other weights are contained in the interior of this parabola (i.e. those points with “=” replaced by “≤”).

The last statement follows because, by translation with an element of W_{aff} we can assume that (h, μ′, m′) is antidominant. Then

|(h, μ′, m′)|^{2} − |(h, μ, m)|^{2} = ⟨(h, μ′, m′) + (h, μ, m), (h, μ′, m′) − (h, μ, m)⟩ ≤ 0,

because the first entry is antidominant and the second one is positive, (h, μ, m) being a lowest weight.

We use the fact (not proved here) that the extension inner product on 𝔱 extends to an inner product on ℝ ⊕ 𝔱 ⊕ ℝ which implements the pairing between roots and coroots.

7 Proof for Section 5 and Homogeneous spaces of LG

We now want to indicate the proofs of the statements of Section 6.

The basic idea is that we can mimic the we can mimic the Borel-Weil theorem for compact Lie groups. It can be stated as follows:

7.1 Theorem. The homogeneous space G/T has a complex structure, because it is isomorphic to G_{ℂ}/B^{+}, where G_{ℂ} is the complexification of G and B^{+} is the Borel subgroup. In case G = U(n), G_{ℂ} = Gl(n, ℂ) and B^{+} ⊂ Gl(n, ℂ) is the subgroup of upper triangular matrices; the homogeneous space is the flag variety.

To each weight λ: T → S^{1} there is a uniquely associated holomorphic line bundle L_{λ} over G_{ℂ}/B^{+} with action of G_{ℂ}.

L_{λ} has non-trivial holomorphic sections if and only if λ is an antidominant weight. In this case, the space of holomorphic section is an irreducible representation with lowest weight λ.

For loop groups, the relevant homogeneous space is

Y := LG/T = LG_{ℂ}/B^{+}G_{ℂ}, B^{+}G_{ℂ} = {∑_{k≥0} λ_{k}z^{k} | λ_{0} ∈ B^{+}}

Recall that for GL(n, ℂ), the Borel subgroup B^{+} is the subgroup of upper triangular matrices.

Note that the second description defines on Y the structure of a complex manifold.

We have also Y = L̃G/T̃ = L̃G_{ℂ}/B̃^{+}G_{ℂ}.

7.2 Lemma. Each character λ: T̃ → S^{1} has a unique extension B̃^{+}G_{ℂ} = T̃_{ℂ} · Ñ^{+}G_{ℂ} → ℂ^{x},, with N^{+}G_{ℂ} := {∑_{k≥0} λ_{k}z^{k} | λ_{0} ∈ N^{+}G_{ℂ}}, where N^{+} is the nilpotent subgroup of G_{ℂ} whose Lie algebra is generated by the positive root vectors, for Gl(n, ℂ) it is the group of upper triangular matrices with 1s on the diagonal. This way defines a holomorphic line bundle

L_{λ} := L̃G_{ℂ} ×_{B̃^{+}G_{ℂ}} ℂ over Y.

Write Γ_{λ} for the space of holomorphic sections of Lλ. This is a representation of L̃G ⋊ S^{1}.

7.3 Lemma. The space Y = LG/T contains the affine Weyl group W_{aff} = (Hom(S^{1}, T) · N(T))/T, where N(T) is the normalizer of T in G ⊂ LG.

Y is stratified by the orbits of W_{aff} under the action of N^{−}LG_{ℂ} := {∑_{k≤0} λ_{k}z^{k} | λ_{0} ∈ N^{−}G_{ℂ}}. Here N^{−}G_{ℂ} is the nilpotent Lie subgroup whose Lie algebra is spanned by the negative root vectors of 𝔤_{ℂ}. For Gl(n, ℂ) this is the group of lower triangular matrices with 1s on the diagonal.

7.4 Theorem. Assume that λ ∈ Hom(T̃, S^{1}) is a weight such that the space Γ_{λ} of holomorphic sections of L_{λ} is non-trivial. Then

(1) Γ_{λ} is a representation of positive energy.

(2) Γ_{λ} is of finite type, i.e. each fixed energy subspace Γ_{λ}(n) is finite dimensional

(3) λ is the lowest weight of Γ_{λ} and is antidominant.

(4) Γ_{λ} is irreducible.

Proof. We use the stratification of Y to reduce to the top stratum. It turns out that the top stratum is N^{−}LG_{ℂ} and that L_{γ} trivializes here, so Γ_{λ} restricts to the space of holomorphic functions on N^{−}LG_{ℂ}. Because holomorphic sections are determined by their values on the top stratum, this restriction map is injective. We can further, with the exponential map (which is surjective in this case), pull back to holomorphic functions on N^{−}L𝔤_{ℂ}, and then look at the Taylor coefficients at 0.

This way we finally map invectively into ∏_{p≥0} S^{p}(N^{−}L𝔤_{ℂ})^{∗}, where S^{p}(V)^{∗} is the space of p-multilinear maps V × ⋯ × V → ℂ. This map is indeed T̃ × S^{1}- equivariant, if we multiply the obvious action on the target with λ.

It now turns out that N^{−}L𝔤_{ℂ} has (essentially by definition) negative energy, and therefore the duals S^{p}(N^{−}L𝔤_{ℂ}) all have positive energy; and the weights are exactly the positive roots (before multiplication with λ). Consequently λ is of lowest weight. If for some positive root (α, n), we had λ(α, n) = m > 0, then reflection in W_{aff} corresponding to α would map λ to λ − mα, which on the other hand can not be a root of Γ_{λ} if λ is a root. Consequently, λ is antidominant.

Explicit calculations also show that the image of the “restriction map” is contained in a subspace of finite type.

To prove that Γ_{λ} is irreducible, we look at the subspace of lowest energy. This is a representation of G_{ℂ}. Pick a lowest weight vector for this representation, it is then invariant under the nilpotent subgroup N^{−}. Since it is of lowest energy, it is even invariant under N^{−}LG_{ℂ}. On the other hand, since the top stratum of Y is N^{−}LG_{ℂ}, the value of an invariant section at one point completely determines it, so that the space of such sections is 1-dimensional. B^{−}LG_{ℂ} acts on this space by multiplication with the holomorphic homomorphism λ: B^{−}LG_{ℂ} → ℂ^{×}, so that λ really occurs as lowest weight.

We now show that this vector is actually a cyclic vector, i.e. generates Γ_{λ} under the action of L̃G ⋊ S^{1}. Else choose a vector of lowest energy not in this subrepresentation, of lowest weight for the corresponding action of the compact group G, and we get with the argument as above a second N^{−}LG_{ℂ} invariant section.

7.5 Lemma. Γ_{λ} ≠ 0 if and only If λ is antidominant.

Proof. If α is a positive root of G ((α, 0 therefore a positive root of LG ⋊ S^{1}), we get a corresponding inclusion i_{α}: Sl_{2}(ℂ) → L̃G_{ℂ} whose restriction to ℂ^{×} ⊂ Sl_{2}(ℂ) is the exponential of h_{α}. Since α is positive, B^{+}Sl_{2}(ℂ) is mapped to B^{+}L̃G_{ℂ}. Therefore, we get an induced map P^{1}(ℂ) = Sl_{2}(ℂ)/B^{+} → Y. The pullback of L_{λ} under this map is the line bundle associated to λ ∘ h_{α}. If Γ_{λ} is non-trivial, we can therefore pull back to obtain a non-trivial holomorphic section of this bundle over P^{1}(ℂ). These exist only if λ ∘ h_{α} = λ(h_{α}) ≤ 0.

This prove half the conditions for antidominance. We omit the prove of the other half, where we have to consider the positive root (−α, 1).

If, on the other hand, λ is antidominant, one constructs a holomorphic section along the stratification of Y.

7.6 Theorem. An arbitrary smooth representation of L̃G⋊S^{1} of positive energy splits (upto essential equivalence) as a direct sum of representations of the form Γ_{λ}.

In particular, the Γ_{λ} are exactly the irreducible representations.

Proof. If E is a representation of positive energy, so is \overline{E}^{∗}. Pick in the Grepresentation \overline{E}^{∗}(0) a vector ϵ of lowest weight λ. For each smooth vector v ∈ E, the map

s_{v}: L̃G_{ℂ} → ℂ; f ↦ ϵ(f^{−1}v)

turns out to define a holomorphic section of L_{λ}. This give a non-trivial map E → Γ_{λ}. If E is irreducible it therefore is essentially equivalent to Γ_{λ}

Using \overline{Γ}_{λ}^{∗} and similar constructions, we can split off factor Γ_{λ} successively from an arbitrary representation E.

The proofs of the statements about the structure of these homogenous spaces uses the study of related Grassmannians. These we define and study for LU(n); results for arbitrary compact Lie groups follow by embedding into U(n) and reduction to the established case.

Let H = H_{+} ⊕ H_{−} be a (polarized) Hilbert space. The important example for us is H = L^{2}(S^{1}, ℂ^{n}), with H_{+} generated by functions z^{k} for k ≥ 0 and H_{−} generated by z^{k} with k < 0 (negative or positive Fourier coefficients vanish).

On this Hilbert space, the complex loop group LGl(n, ℂ) acts by pointwise multiplication.

We define the Grassmannian Gr(H) := {W ⊂ H | pr_{+} : W → H_{+} is Fredholm, pr_{=} : W → H_{−} is Hilbert-Schmidt}. This is a Hilbert manifold.

We define the restricted linear group Gl_{res}(H) := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) | b, c Fredholm\}. (This implies that a, d are Hilbert-Schmidt). Set U_{res}(H) := U(H) ∩ Gl_{res}(H). These groups act on Gr(H).

7.7 Lemma. The image of LGl(n, ℂ) in Gl(H) is contained in Gl_{res}(H).

Proof. Write γ = ∑_{k} γ_{k}e^{ikθ} = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ LGl(n, ℂ). Then for each n > 0,

b(e^{−inθ}) = ∑_{k≥n} γ_{k}e^{i(k−n)θ}, and for n ≥ 0, c(e^{inθ}) = ∑_{k<−n} γ_{k}e^{i(k+n)θ}. Therefore ∥b∥_{HS} = ∑_{n≥0} ∑_{k≥n} |γ_{k}|^{2} = ∑_{k≥0}(1 + k) |γ_{k}|^{2} < ∞ since smooth functions have rapidly decreasing Fourier expansion. Similarly for c.

Gr(H) has a cell decomposition/stratification in analogy to the finite dimensional Grassmannians (with infinitely many cells of all kinds of relative dimensions). Details are omitted here, but this is an important tool in many proofs.

Gr(H) contains the subspace Gr_{∞}(H) := {W ∈ Gr(H) | im(pr_{−})∪im(pr_{+}) ⊂ C^{∞}(S^{1}, ℂ^{n})}. Inside this one we consider the subspace Gr_{∞}^{(n)} := {W ∈ Gr_{∞}(H) | zW ⊂ W }.

It turns out that Gr_{∞}(H)^{(n)} = LGl(n, complexs)/L^{+}Gl(n, ℂ) = LU(n)/U(n). The main point of the definition of Gr(H) is that we can define a useful

(and fine) virtual dimension for its elements; measuring the dimension of W ∩ z^{m}H_{−} for every m. This is used to stratify these Grassmannians and related homogeneous spaces, and these stratifications were used in the study of the holomorphic sections of line bundles over these spaces.

References

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[2] Karl-Hermann Neeb. Borel-Weil theory for loop groups. In Infinite dimensional Kähler manifolds (Oberwolfach, 1995), volume 31 of DMV Sem., pages 179–229. Birkhäuser, Basel, 2001.

[3] Andrew Pressley and Graeme Segal. Loop groups. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1986. Oxford Science Publications. 1