arXiv subjects
Publications and source records attributed to Rudrendra Kashyap.
Let $G$ be a connected reductive complex algebraic group and let $q\in\mathbb{C}^{\times}$ be not a root of unity. We prove that every $(G,q)$-connection on a formal punctured disc admits a $(G,q)$-oper structure.
Let $G$ be a connected complex reductive algebraic group. We study finite-rank flat algebraic connections on trivial vector bundles whose connection forms are left invariant, allowing arbitrary algebraic horizontal morphisms. We prove that a flat algebraic connection on $G$ is regular-singular if and only if its pullback to the canonical finite central cover is isomorphic to a left-invariant flat algebraic connection on a trivial vector bundle. Moreover, every regular-singular algebraic connection on $G$ is a direct summand of such a connection. We characterize the essential image of pullback from the abelianization by the vanishing of a derived-monodromy obstruction and show that pullback is an equivalence precisely when $G^{\mathrm{der}}$ is simply connected. For semisimple $G$, regular-singular connections are classified by finite-dimensional representations of the finite central kernel of the simply connected cover. We also classify regular-singular connections on $\mathrm{GL}_r$ by pullback along the determinant, give a $\mathrm{PGL}_2$ counterexample to the abelianization classification of left-invariant trivial-bundle algebraic connections, and compute de Rham and Betti cohomology in the semisimple case.
Hecke operators on moduli of bundles over a global function field become substantially more complicated in the presence of ramification. We show that far enough in the Harder-Narasimhan cone of $\mathrm{Bun}_G$, this extra complexity has a simple structure, which allows to reduce most of the study to the unramified case. Using the theory of graphs of Hecke operators, we transform this statement into a combinatorial condition. Utilizing the combinatorial language, we obtain tight bounds, and for generic eigenvalues exact formulas for the dimensions of Hecke eigenspaces with arbitrary ramification for $\mathrm{Bun}_{\mathrm{PGL}_2}$. We compare these formulas to the known results in the theory of Eisenstein series. Moreover, our methods allow to construct eigenforms explicitly.
We study Hecke operators on moduli spaces of ramified $G$-bundles using the combinatorial language of Hecke graphs. We introduce a general notion of $\mathcal H$-ramification in the spirit of parahoric ramification, which depends on a choice of a divisor and subgroups of $G$ at every point of the divisor. Building on our previous work, we prove that, under mild regularity conditions, the action of a Hecke operator in the deep cusp of $\mathrm{Bun}_G$ in a highly complex ramification mimics an action in a much simpler ramification. This reduces the study to a smaller number of cases which, in particular, involve divisors supported at no more than two points. We demonstrate our methods by computing various examples for $G=\mathrm{PGL}_2$ and computing the dimensions of spaces of Hecke eigenforms for generic eigenvalues. We connect the obtained dimension formulas with the known results from the theory of Eisenstein series.