Suppose that G and H are connected reductive groups over a number field F and that an L-homomorphism ρ : LG −→ LH is given. The Langlands functoriality conjecture predicts the existence of a map from the automorphic representations of G(A) to those of H(A). If the adelic points of the algebraic groups G, H are replaced by their metaplectic covers, one may hope to specify an analogue of the Lgro...

The representation theory of reductive groups, such as the group GLn of invertible complex matrices, is an important topic, with applications to number theory, algebraic geometry, mathematical physics, and quantum topology. One way to study this representation theory is through the geometric Satake correspondence (also known as geometric Langlands duality). This correspondence relates the geome...

We establish integral analogues of results of Bushnell and Henniart [BH] for spaces of Whittaker functions arising from the groups GLn(F ) for F a p-adic field. We apply the resulting theory to the existence of representations arising from the conjectural “local Langlands correspondence in families” of [EH], and reduce the question of the existence of such representations to a natural conjectur...

We introduce a new approach to the representation theory of reductive p-adic groups G, based on the Geometric Invariant Theory (GIT) of Moy-Prasad quotients. Stable functionals on these quotients are used to give a new construction of supercuspidal representations of G having small positive depth, called epipelagic. With some restrictions on p, we classify the stable and semistable functionals ...

We prove that any flat G-bundle, where G is a complex connected reductive algebraic group, on the punctured disc admits the structure of an oper. This result is important in the local geometric Langlands correspondence proposed in [7]. Our proof uses certain deformations of the affine Springer fibers which could be of independent interest. As a byproduct, we construct representations of affine ...

We give cases in which nearby cycles commute with pushforward from sheaves on the moduli stack of shtukas to a product curves over finite field. The proof systematically uses property that taking Satake Beilinson–Drinfeld Grassmannian parahoric reduction is central functor together ‘Zorro's lemma’ argument similar Xue [ Smoothness cohomology stacks , Preprint (2020), arXiv:2012.12833]. As an ap...