Let X be a curve over F_q and let G be a reductive group. The classical theory of automorphic forms is concerned with the space of functions on the quotient G_A/G_κ, where κ(resp., A) is the field or rational functions on X (resp., the ring of adeles of κ). In this paper, we will consider only the unratified situation, i.e., we will study functions (and afterwards perverse sheaves) on the double quotient G_OG_A/G_κ.
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