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Pursuing the double affine Grassmannian. I: Transversal slices via instantons on \(A_k\)-singularities. (English) Zbl 1200.14083
Let \(G\) be a reductive group and \(G^{\vee}\) be its Langlands dual. The geometric Satake isomorphism is an identification of the category of finite dimensional representations of \(G^{\vee}\) with the category of equivariant perverse sheaves on the affine Grassmannian of \(G\). Under this isomorphism, simple \(G^{\vee}\)-modules can be identified with certain intersection cohomology groups of varieties that are obtained from orbits of the affine Grassmannian. This relationship allows one to compute certain intersection cohomology groups via the representation theory of \(G^{\vee}\). The goal of this paper is to begin to develop a comparable theory for affine Kac-Moody groups which will be continued in forthcoming work.
For \(G\) semisimple and simply connected, let \(G_{\text{aff}}\) denote the corresponding (untwisted) affine Kac-Moody group and \(G_{\text{aff}}^{\vee}\) denotes its dual. The idea presented in the paper is that the integrable representations of \(G_{\text{aff}}^{\vee}\) should be related to the geometry of certain moduli spaces of \(G\)-bundles on affine two-space. The authors nicely describe the intuition behind the program and then introduce precisely the varieties and \(G\)-bundles of interest. They make a number of conjectures about how things should behave, with the main conjecture involving intersection cohomology. The conjectures are verified for representations of level \(k\) where \(k\) is sufficiently large. Slightly weaker results are obtained for level one representations for arbitrary \(G\) and for arbitrary levels when \(G\) is the special linear group.

MSC:
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14M15 Grassmannians, Schubert varieties, flag manifolds
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
20G44 Kac-Moody groups
22E57 Geometric Langlands program: representation-theoretic aspects
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