tailieunhanh - Đề tài " 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 co-roots appear in a symmetric manner and so the connected reductive algebraic groups come ˇ in pairs. | Annals of Mathematics Geometric Langlands duality and representations of algebraic groups over commutative rings By I. Mirkovi c andK. Vilonen Annals of Mathematics 166 2007 95 143 Geometric Langlands duality and representations of algebraic groups over commutative rings By I. MiRKOVic and K. ViLONEN 1. Introduction 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 co-roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs. If G is a reductive group we write G 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 cornerstone of the Langlands program. It also appeared later in physics MO GNO . In this paper we discuss the basic relationship between G and G. We begin with a reductive G and consider the affine Grassmannian Qr the Grassmannian for the loop group of G. For technical reasons 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 Qr. 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 G. In this interpretation the spherical perverse sheaves on the affine Grassmannian correspond to finite dimensional complex representations of G. Thus instead of defining G in terms of the classification of reductive groups we provide a canonical construction of G starting from G. We can carry out our construction over the integers. The .