Geometrizing the minimal representations of even orthogonal groups

TL;DR

This paper provides a geometric interpretation of automorphic functions related to minimal representations of even orthogonal groups by constructing a perverse sheaf on the moduli stack of SO_{2n}-torsors, connecting automorphic forms with geometric objects.

Contribution

It introduces a new geometric construction of automorphic functions for SO_{2n} using perverse sheaves and geometric theta-lifting, with explicit calculations for low-genus curves.

Findings

Constructed a perverse sheaf K on Bun_{SO_{2n}} representing the automorphic function.

Explicitly calculated K for genus zero and one curves.

Formulated conjectures extending the construction to more general algebraic groups.

Abstract

Let X be a smooth projective curve. Write Bun_{SO_{2n}} for the moduli stack of SO_{2n}-torsors on X. We give a geometric interpretation of the automorphic function f on Bun_{SO_{2n}} corresponding to the minimal representation. Namely, we construct a perverse sheaf K on Bun_{SO_{2n}} such that f should be equal to the trace of Frobenius of K plus some constant function. We also calculate K explicitely for curves of genus zero and one. The construction of K is based on some explicit geometric formulas for the Fourier coefficients of f on one hand, and on the geometric theta-lifting on the other hand. Our construction makes sense for more general simple algebraic groups, we formulate the corresponding conjectures. They could provide a geometric interpretation of some unipotent automorphic representations in the framework of the geometric Langlands program.