  We study automorphic categories of nilpotent sheaves under degenerations of
smooth curves to nodal Deligne-Mumford curves. Our constructions realize affine
Hecke operators as the result of bubbling projective lines from marked points.
We use this to construct a "gluing functor" from the automorphic category of a
nodal Deligne-Mumford curve to the automorphic category of a smoothing.
