The aim of these notes is to generalize Laumon’s construction [20] of automorphic sheaves corresponding to local systems on a smooth, projective curve $C$ to the case of local systems with indecomposable unipotent ramification at a finite set of points. To this end we need an extension of the notion of parabolic structure on vector bundles to coherent sheaves. Once we have defined this, a lot of arguments from the article “ On the geometric Langlands conjecture” by Frenkel, Gaitsgory and Vilonen [11]...

Moduli spaces of vector bundles on families of non-singular curves are usually compactified by considering (slope)semistable bundles on stable curves. Alternatively, one could consider Hilbert-stable curves in Grassmannians. We study some properties of the latter and compare them with similar properties of curves coming from the former compactification. This leads to a new interpretation of the moduli space of (semi)stable torsion-free sheaves on a fixed nodal curve. One can present it as a quotient...

We prove that the global geometric theta-lifting functor for the dual pair $(H,G)$ is compatible with the Whittaker functors, where $(H,G)$ is one of the pairs $({\mathrm{S}𝕆}_{2n},{𝕊p}_{2n})$, $({𝕊p}_{2n},{\mathrm{S}𝕆}_{2n+2})$ or $({𝔾\mathrm{L}}_n,{𝔾\mathrm{L}}_{n+1})$. That is, the composition of the theta-lifting functor from $H$ to $G$ with the Whittaker functor for $G$ is isomorphic to the Whittaker functor for $H$.

We give a characterization of conformal blocks in terms of the singular cohomology of suitable smooth projective varieties, in genus $0$ for classical Lie algebras and $G_2$.