Geometrizing the minimal representations of even orthogonal groups

Let X \mathrm {Bun}_{\mathrm {S}\mathbb{O}_{2n}} \mathrm {S}\mathbb{O}_{2n}. We give a geometric interpretation of the automorphic function f \mathrm {Bun}_{\mathrm {S}\mathbb{O}_{2n}} \mathcal {K}_H \mathrm {Bun}_{\mathrm {S}\mathbb{O}_{2n}} should be equal to the trace of the Frobenius of \mathcal {K}_H 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.