Geometric Langlands for hypergeometric sheaves

Abstract

Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler-Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. In this paper, we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems, thus confirming a central conjecture of the geometric Langlands program for hypergeometrics. The key new concept is the notion of hypergeometric automorphic data. We prove that this automorphic data is generically rigid (in the sense of Zhiwei Yun) and identify the resulting Hecke eigenvalue with hypergeometric sheaves. The definition of hypergeometric automorphic data in the tame case involves the mirabolic subgroup, while in the wild case, semistable (but not necessarily stable) vectors coming from principal gradings intervene.

Rigidity method for automorphic forms over function fields

“…Example (Hypergeometric automorphic data). In [24], Kamgarpour and Yi constructed automorphic data realizing Katz's hypergeometric local systems. Let G = PGL n , X = P 1 .…”

“…For a specific choice of the partial flag V • where the dimensions of V i are distributed as evenly as possible, it was proved in [24] that (K S , K S ) is geometrically rigid.…”