Choose $q\in ℂ$ with $0\<\left|q\right|\<1$. The main theme of this paper is the study of linear $q$-difference equations over the field $K$ of germs of meromorphic functions at $0$. A systematic treatment of classification and moduli is developed. It turns out that a difference module $M$ over $K$ induces in a functorial way a vector bundle $v\left(M\right)$ on the Tate curve $E_q:={ℂ}^{*}/q^{\mathbb{Z}}$ that was known for modules with ”integer slopes“, [Saul, 2]). As a corollary one rediscovers Atiyah’s classification $\left(\right[\mathrm{A}t\left]\right)$ of the indecomposable vector bundles on the complex Tate...

We develop a theory of general sheaves over weighted projective lines. We define and study a canonical decomposition, analogous to Kac's canonical decomposition for representations of quivers, study subsheaves of a general sheaf, general ranks of morphisms, and prove analogues of Schofield's results on general representations of quivers. Using these, we give a recursive algorithm for computing properties of general sheaves. Many of our results are proved in a more abstract setting, involving a hereditary...

Let $X$ be a smooth projective curve over an algebraically closed field of characteristic $\>2$. Consider the dual pair $H={\mathrm{SO}}_{2m},G={\mathrm{Sp}}_{2n}$ over $X$ with $H$ split. Write ${\mathrm{Bun}}_G$ and ${\mathrm{Bun}}_H$ for the stacks of $G$-torsors and $H$-torsors on $X$. The theta-kernel ${\mathrm{Aut}}_{G,H}$ on ${\mathrm{Bun}}_G\times {\mathrm{Bun}}_H$ yields theta-lifting functors $F_G:\mathrm{D}\left({\mathrm{Bun}}_H\right)\to \mathrm{D}\left({\mathrm{Bun}}_G\right)$ and $F_H:\mathrm{D}\left({\mathrm{Bun}}_G\right)\to \mathrm{D}\left({\mathrm{Bun}}_H\right)$ between the corresponding derived categories. We describe the relation of these functors with Hecke operators. In two particular cases these functors realize the geometric Langlands functoriality for the above pair (in the non ramified case)....