Geometrizing the minimal representations of even orthogonal groups

Abstract

Let X X be a smooth projective curve. Write B u n S O 2 n \mathrm {Bun}_{\mathrm {S}\mathbb {O}_{2n}} for the moduli stack of S O 2 n \mathrm {S}\mathbb {O}_{2n} -torsors on X X . We give a geometric interpretation of the automorphic function f f on B u n S O 2 n \mathrm {Bun}_{\mathrm {S}\mathbb {O}_{2n}} corresponding to the minimal representation. Namely, we construct a perverse sheaf K H \mathcal {K}_H on B u n S O 2 n \mathrm {Bun}_{\mathrm {S}\mathbb {O}_{2n}} such that f f should be equal to the trace of the Frobenius of K H \mathcal {K}_H plus some constant function. The construction is based on some explicit geometric formulas for the Fourier coefficients of f 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.