Braverman, Alexander; Finkelberg, Michael Pursuing the double affine Grassmannian. II: Convolution. (English) Zbl 1242.14047 Adv. Math. 230, No. 1, 414-432 (2012). Let \(G\) be a reductive group and \(G^{\vee}\) denote its Langlands dual. The categorical version of the Satake equivalence is an identification as tensor categories of the category of finite dimensional representations of \(G^{\vee}\) with the category of equivariant perverse sheaves on the affine Grassmannian \(\text{Gr}_{G}\) 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.For \(G\) semisimple and simply connected, let \(G_{\text{aff}}\) denote the corresponding affine Kac-Moody group and \(G_{\text{aff}}^{\vee}\) denotes its dual. This paper is a continuation of previous work of the authors [Duke Math. J. 152, No. 2, 175–206 (2010; Zbl 1200.14083)]. Their goal is to construct an analogue of \(\text{Gr}_{G}\) associated to \(G_{\text{aff}}\) which would have a similar connection with the representation theory of \(G_{\text{aff}}^{\vee}\). This is the so-called double affine Grassmannian.The focus here is on the convolution which gives rise to the tensor structure on the aforementioned category of perverse sheaves. In particular, for any integer \(n \geq 2\), there is an \(n\)-fold convolution map \(\text{Gr}_{G}\star \cdots \star \text{Gr}_{G} \to \text{Gr}_{G}\). The authors construct certain Uhlenbeck spaces and a proper birational morphism between them which the authors conjecture to be the correct analogue of the restriction of the \(n\)-fold convolution map to orbital subvarieties. This construction makes use of quiver varieties and work of H. Nakajima [SIGMA 5, Paper 003, 37 p. (2009; Zbl 1241.17028)] for the special linear group. The authors conjecture that the multiplicity of a simple \(G_{\text{aff}}^{\vee}\)-module in a certain tensor product of simple modules can be identified with the multiplicity of an intersection cohomology group in the image under the aforementioned morphism of another intersection cohomology group. Using the work of Nakajima, this conjecture is verified for the special linear group.The authors also include a discussion of how these constructions would look in a double affine version of the Beilinson-Drinfeld Grassmannian. Reviewer: Christopher P. Bendel (Menomonie) Cited in 1 ReviewCited in 7 Documents MSC: 14M15 Grassmannians, Schubert varieties, flag manifolds 14D20 Algebraic moduli problems, moduli of vector bundles 14D23 Stacks and moduli problems 14D24 Geometric Langlands program (algebro-geometric aspects) 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14M27 Compactifications; symmetric and spherical varieties 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 20G44 Kac-Moody groups 22E57 Geometric Langlands program: representation-theoretic aspects Keywords:affine Grassmannian; convolution; quiver varieties; reductive group; Kac-Moody group; Langlands duality; intersection cohomology; moduli space; Uhlenbeck space; Beilinson-Drinfeld Grassmannian; Kleinian stacks Citations:Zbl 1200.14083; Zbl 1241.17028 PDFBibTeX XMLCite \textit{A. 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