Abstract. We prove a conjecture of Drinfeld regarding restriction of central extensions of an algebraic group G to the commutator subgroup. As an application, we construct the "true commutator" of G. The quotient of G by the action of the true commutator is the universal commutative group stack to which G maps.

Let G be a parahoric group scheme over a complex projective curve X of genus greater than one. Let BunG denote the moduli stack of G-torsors on X. We prove several results concerning the Hitchin map on T * BunG. We first show that the parahoric analogue of the global nilpotent cone is isotropic and use this to prove that BunG is "very good" in the sense of Beilinson-Drinfeld. We then prove that the parahoric Hitchin map is a Poisson map whose generic fibres are abelian varieties. Together, these results imply that the parahoric Hitchin map is a completely integrable system.

Abstract. We determine the image of the (strongly) parabolic Hitchin map for all parabolics in classical groups and G2. Surprisingly, we find that the image is isomorphic to an affine space in all cases, except for certain "bad parabolics" in type D, where the image can be singular.

Abstract. It is expected that, under mild conditions, local Langlands correspondence preserves depths of representations. In this article, we formulate a conjectural geometrisation of this expectation. We prove half of this conjecture by showing that the depth of a categorical representation of the loop group is less than or equal to the depth of its underlying geometric Langlands parameter. A key ingredient of our proof is a new definition of the slope of a meromorphic connection, a definition which uses opers. In the appendix, we consider a relationship between our conjecture and Zhu's conjecture on non-vanishing of the Hecke eigensheaves produced by Beilinson and Drinfeld's quantisation of Hitchin's fibration for non-constant groups.

We define the p-adic trace of certain rank-one local systems on the multiplicative group over p-adic numbers, using Sekiguchi and Suwa's unification of Kummer and Artin-Schreier-Witt theories. Our main observation is that, for every nonnegative integer n, the p-adic trace defines an isomorphism of abelian groups between local systems whose order divides ( p − 1) p n and -adic characters of the multiplicative group of p-adic integers of depth less than or equal to n.