Preservation of depth in the local geometric Langlands correspondence

Abstract

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.

Classification of Argyres-Douglas theories from M5 branes

“…The ratio s = a/b (we take b to be the minimal possible integer possible) is called the slope or Katz invariant associated with the local model of the Higgs field [46,47]. It can be shown using the relation between Higgs bundles and opers that the denominator b of s must always be a divisor of d i for some 1 ≤ i ≤ rank (J), which are degrees of the fundamental invariants of J (see Table 3) [47]. Further imposing that T is regular semisimple restricts the denominator b of the slope to take the values summarized in Table 1.…”

On the Notion of Conductor in the Local Geometric Langlands Correspondence

“…Frenkel and Gaitsgory have extensively analysed the local geometric Langlands correspondence when the underlying connection is regular singular (i.e., its irregularity equals zero). Previously, we have examined some of the features of the irregular case [Kam14], [CK14], [KS15]. This paper is an attempt to understand yet another aspect of the local geometric Langlands correspondence in the presence of irregular connections.…”

Depth and the Local Langlands Correspondence

“…The following result may be considered as the non-archimedean analogue of [ChKa,Theorem 1] in the case of the geometric local Langlands correspondence.…”