The module structure of the equivariant K-theory of the based loop group of SU(2)

Let G be a compact connected Lie group. The G-equivariant topological K-theory K * G (X) of a topological G-space X is an object of intrinsic interest, carrying information about X which reflects the G-action on X. The space G itself, with G acting by conjugation, and its space of (continuous) based loops ΩG with the induced (pointwise) action, are two examples of natural and important G-spaces. For Lie groups G, the ordinary and Borel-equivariant cohomology rings H * (G), H * (ΩG), H * G (G), and H * G (ΩG) were computed decades ago (with contributions from many people), and these results are by now considered classical; the same is true of the computations of the ordinary Ktheory rings K * (G) and K * (ΩG). A brief account of some of these 'classical' results is contained in Section 2. However, computing the equivariant K-theory of these spaces proved to be more difficult. For instance, K * G (G) was only recently computed by Brylinski and Zhang in 2000 [8]. The chief contribution of this manuscript is a concrete computation of the module structure of K * G (ΩG) for the specific case G = SU (2). In addition to being of basic interest, this computation is also motivated by questions from symplectic geometry which we briefly describe at the end of this introduction. For now we note that, at the beginning of work on this manuscript, our goal was a full and explicit computation of both the module and product structures of K * G (ΩG) when G = SU (2).

The present manuscript describes the module structure, while the product structure is computed in the companion paper ( [11]). We now proceed to briefly describe our results and methods. We view K * G (ΩG) as a module over K * G = K * G (pt) ∼ = R(G). For the following let Ω poly G denote the subspace of polynomial loops in G, and Ω poly,r G the subspace of polynomial loops of degree ≤ r. For a precise definition we refer the reader to equation 3.1. (The spaces Ω poly,r G form a filtration of Ω poly G.) With this notation in place we may state the main theorem of this manuscript (Theorem 6.5), which asserts in particular that K * G (ΩG) (respectively K * T (ΩG)) is an inverse limit of free R(G)-modules (respectively R(T )-modules). Theorem 1.1. Let G = SU (2) and let T denote its maximal torus. Let ΩG denote the space of based loops in G, equipped with the pointwise conjugation action of G. The R(G)-module (respectively R(T )-module) K * G (ΩG) (respectively K * T (ΩG)) can be described as follows:

This theorem should not be surprising to experts for two reasons. Firstly, the computation for the non-equivariant case, using an analogous filtration, follows from the work of many other authors: for instance, James [16] described in 1955 a filtration of spaces of the form ΩΣX, which applies to our situation of G = SU (2) since SU (2) ∼ = S 3 ∼ = ΣS 2 , while Pressley and Segal develop a theory for general loop groups ΩG in [24], which was further developed by Mitchell in [23]. Indeed, the technical geometric tools for our argument are G-equivariant analogues of the ideas of Pressley and Segal. However, our geometric results are not immediate corollaries of those in [24], mainly due to Theorem 5.7. The non-equivariant analogue of Theorem 5.7 in [24] is a description of a certain space as a product of contractible spaces [24, (8.4.4)], while in Theorem 5.7, we instead get a nontrivial bundle over P 1 . This geometric distinction is relevant in our analysis. Secondly, statements similar to Theorem 1.1 for the (T × S 1 )-equivariant K-theory K * T ×S 1 (ΩG) can be deduced by Kac-Moody methods (see ) or GKM methods (see e.g. Harada-Henriques-Holm [10]). However, our G-equivariant result is not an immediate corollary of these torus-equivariant results since, for instance, it is not always the case for a G-space X that K * G (X) ∼ = K * T (X) W (cf. for example [13,Example 4.8]), where W is the Weyl group. For this reason we worked instead with G-equivariant analogues of the approach in [24]. (In fact, as it turns out, direct computation confirms that the isomorphism K * G (ΩG) ∼ = K * T (ΩG) W is satisfied in our case.) We now summarize the strategy of our computation in some more detail. Let Ω poly G and Ω psm G denote the subspaces of polynomial and piecewise smooth loops, respectively, in ΩG. (Both are defined more precisely below.) One of our key steps is to prove that there are G-equivariant homotopy equivalences (1.1)

This reduces our computation to that of K * G (Ω poly G). Our second essential strategy is to analyze the G-filtration of Ω poly G by the spaces Ω poly,r G for r ∈ Z >0 , consisting of loops of polynomial degree ≤ r. More specifically, we prove that the filtration quotients (1.2) Ω poly,r G/Ω poly,r-1 G are G-homeomorphic to Thom spaces of complex G-vector bundles over P 1 , implying that their equivariant K-theory can be computed via the (equivariant) Thom isomorphism theorem. From this, a computation of K G (Ω poly G) is obtained by induction and ta

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