The Tannakian formalism and the Langlands conjectures (original) (raw)

Semisimplification for Subgroups of Reductive Algebraic Groups

2020

Let G be a reductive algebraic group—possibly non-connected—over a field k and let H be a subgroup of G. If G= GL_n then there is a degeneration process for obtaining from H a completely reducible subgroup H' of G; one takes a limit of H along a cocharacter of G in an appropriate sense. We generalise this idea to arbitrary reductive G using the notion of G-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a G-completely reducible subgroup H' of G, unique up to G(k)-conjugacy, which we call a k-semisimplification of H. This gives a single unifying construction which extends various special cases in the literature (in particular, it agrees with the usual notion for G= GL_n and with Serre's "G-analogue" of semisimplification for subgroups of G(k)). We also show that under some extra hypotheses, one can pick H' in a more canonical way using the Tits Cen...

A Tannakian Context for Galois

arXiv (Cornell University), 2011

Strong similarities have been long observed between the Galois (Categories Galoisiennes) and the Tannaka (Categories Tannakiennes) theories of representation of groups. In this paper we construct an explicit (neutral) Tannakian context for the Galois theory of atomic topoi, and prove the equivalence between its fundamental theorems. Since the theorem is known for the Galois context, this yields, in particular, a proof of the fundamental (recognition) theorem for a new Tannakian context. This example is different from the additive cases or their generalization, where the theorem is known to hold, and where the unit of the tensor product is always an object of finite presentation, which is not the case in our context.

Two contributions to the representation theory of algebraic groups

2002

Sei V ein endlich-dimensionaler, komplexer Vektorraum. Eine Teilmenge X in V hat die Trennungseigenschaft, falls das Folgende gilt: Für je zwei linear unabhängige lineare Funktionen l, m auf V existiert ein Punkt x in X mit l(x) = 0 und m(x) = 0. Wir interessieren uns für den Fall V = C[x, y] n , d.h. V ist eine irreduzible Darstellung von SL 2 . Die Teilmengen, die wir untersuchen, sind Bahnabschlüsse von Elementen aus C[x, y] n . Wir beschreiben die Bahnen, die die Trennungseigenschaft erfüllen:

A Tannakian context for Galois theory

Advances in Mathematics, 2013

Strong similarities have been long observed between the Galois (Categories Galoisiennes) and the Tannaka (Categories Tannakiennes) theories of representation of groups. In this paper we construct an explicit (neutral) Tannakian context for the Galois theory of atomic topoi, and prove the equivalence between its fundamental theorems. Since the theorem is known for the Galois context, this yields, in particular, a proof of the fundamental (recognition) theorem for a new Tannakian context. This example is different from the additive cases or their generalization, where the theorem is known to hold, and where the unit of the tensor product is always an object of finite presentation, which is not the case in our context.

Kloosterman sheaves for reductive groups

Annals of Mathematics, 2013

Deligne constructed a remarkable local system on È 1 − {0, ∞} attached to a family of Kloosterman sums. Katz calculated its monodromy and asked whether there are Kloosterman sheaves for general reductive groups and which automorphic forms should be attached to these local systems under the Langlands correspondence.

Algebraic groups over a 2-dimensional local field: Irreducibility of certain induced representations

Journal of Differential Geometry - J DIFFEREN GEOM, 2005

Let G be a split reductive group over a local field K, and let G((t)) be the corresponding loop group. In [1], we have introduced the notion of a representation of (the group of K-points) of G((t)) on a pro-vector space. In addition, we have defined an induction procedure, which produced G((t))-representations from usual smooth representations of G. We have conjectured that the induction of a cuspidal irreducible representation of G is irreducible. In this paper, we prove this conjecture for G=SL<sub>2</sub>.

Tannakization in derived algebraic geometry

Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2014

In this paper we begin studying tannakian constructions in ∞-categories and combine them with the theory of motivic categories developed by Hanamura, Levine, and Voevodsky. This paper is the first in a series of papers. For the purposes above, we first construct a derived affine group scheme and its representation category from a symmetric monoidal ∞-category, which we shall call the tannakization of a symmetric monoidal ∞-category. It can be viewed as an ∞-categorical generalization of work of Joyal-Street and Nori. Next we apply it to the stable ∞-category of mixed motives equipped with the realization functor of a mixed Weil cohomology. We construct a derived motivic Galois group which represents the automorphism group of the realization functor, and whose representation category satisfies an appropriate universal property. As a consequence, we construct an underived motivic Galois group of mixed motives, which is a pro-algebraic group and has nice properties. Also, we present ba...

A Note on the Automorphic Langlands Group

Canadian Mathematical Bulletin, 2002

Langlands has conjectured the existence of a universal group, an extension of the absolute Galois group, which would play a fundamental role in the classification of automorphic representations. We shall describe a possible candidate for this group. We shall also describe a possible candidate for the complexification of Grothendieck's motivic Galois group.