Abstract: Given a group G and marked genus zero curve over a finite field, consider the space of automorphic functions with Iwahori level structure. This is the space of functions on the moduli of parabolic G bundles. I’ll explain the action of Hecke operators by geometric correspondences and give a conjecture about the subspace of pseudo-Eisenstein series. For zero, one, or two markings, this conjecture recovers the known fact that the space of automorphic functions is: (0) the regular module of spherical Hecke algebra, (1) the anti-spherical module of affine Hecke algebra, or (2) the regular bimodule of affine Hecke algebra.

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