  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.
