It is expected that, under mild conditions, the 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 greater 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.
Bibliographical noteFunding Information:
We would like to thank C. Bremer, A. Molev, D. Sage, Z. Yun and X. Zhu for helpful conversations. The first author learned the definition of slope via opers, which is crucial in this paper, from X. Zhu. He is happy to thank him. The second author was supported by the Australian Research Council Discovery Early Career Research Award.
© 2016 American Mathematical Society.
- Affine vertex algebras
- Local geometric Langlands
- Moy-Prasad theory
- Segal-Sugwara operators
- Slope of connections