The constant sheaf on Bun_G

Given a smooth projective curve X and a reductive group G, the geometric Langlands equivalence proved by Gaitsgory, Raskin et al. (roughly) gives an equivalence between sheaves on the stack Bun_G(X) of G-bundles on X (automorphic side) and quasi-coherent sheaves on the stack of G^-local systems on X (spectral side). To compute the image of an object under the geometric Langlands equivalence, one usually bootstraps from the Whittaker model. This method fails for the constant sheaf on Bun_G(X), which is “maximally singular.” Still, we will compute its image under the equivalence, confirming a conjecture of V. Lafforgue. As a consequence, when X is over F_q we find a spectral description for the constant function on Bun_G(X)(F_q