Geometric Langlands Duality and Representations of Algebraic Groups over Commutative Rings

In this paper we give a geometric version of the Satake isomorphism [Sat]. As such, it can be viewed as a first step in the geometric Langlands program. The connected complex reductive groups have a combinatorial classification by their root data. In the root datum the roots and the coroots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs. If G is a reductive group, we write Ǧ for its companion and call it the dual group G. The notion of the dual group itself does not appear in Satake’s paper, but was introduced by Langlands, together with its various elaborations, in [L1, L2] and is a corner stone of the Langlands program. It also appeared later in physics [MO, GNO]. In this paper we discuss the basic relationship between G and Ǧ. We begin with a reductive G and consider the affine Grassmannian Gr, the Grassmannian for the loop group of G. For technical reason we work with formal algebraic loops. The affine Grassmannian is an infinite dimensional complex space. We consider a certain category of sheaves, the spherical perverse sheaves, on Gr. These sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a Hopf algebra, by what has come to be known as Tannakian formalism. The resulting Hopf algebra turns out to be the ring of functions on Ǧ. In this interpretation, the spherical perverse sheaves on the affine Grassmannian correspond to finite dimensional complex representations of Ǧ. Thus, instead of defining Ǧ in terms of the classification of reductive groups, we provide a canonical construction of Ǧ, starting from G. We can carry out our construction over the integers. The spherical perverse sheaves are then those with integral coefficients, but the Grassmannian remains a complex algebraic object. The resulting Ǧ turns out to be the Chevalley scheme over the integers, i.e., the unique split reductive group scheme whose root datum coincides with that of the complex Ǧ. Thus, our result can also be viewed as providing an explicit construction of the Chevalley scheme. Once we have a construction over the integers, we have one for every commutative ring and in particular for all fields. This provides another way of viewing our result: it provides a geometric interpretation of representation theory of algebraic groups over arbitrary rings. The change of rings on the representation theoretic side corresponds to change of coefficients of perverse sheaves, familiar from the universal coefficient theorem in algebraic topology. Note that for us it is crucial that we first prove our result for the integers (or p-adic integers) and then deduce the