Tannakian Formalism Research Articles

Geometric Langlands duality and representations of algebraic groups over commutative rings

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 [LI], [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 Qx, the Grassmannian for the loop group of G. For technical reasons we work with formal algebraic loops. The affine Grassmannian is an infinite dimen sional complex space. We consider a certain category of sheaves, the spherical perverse sheaves, on ?r. 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 Grassman nian correspond to finite dimensional complex representations of G. Thus, instead of defining G in terms of the classification of reductive groups, we pro vide a canonical construction of G, 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.

Some results on multiplicities for SL(n)

We define the multiplicity and the global multiplicity of anL-packet of SL(n), unifying lack of multiplicity one and non-rigidity ofL-packets. The first examples of these phenomena were given by Blasius. Giving a heuristic approach to its calculation, based on Langlands’ Tannakian formalism, we conjecture that the global multiplicity is bounded in terms ofn only. We justify the heuristics in a special case ofL-packets attached to Hecke characters on an Abelian orp-extension. We then focus onL-packets lifted from endoscopic tori. A full description of their global multiplicities is given in the case wheren is prime.