Hecke operators on ramified G-bundles in complex ramification mimic simpler cases under mild conditions, reducing the problem to divisors at most two points and allowing dimension computations for PGL_2 eigenforms.
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For the standard representation of Sp_{2n}(C), the Gaiotto locus is the Bialynicki-Birula closure associated to U(Sp_{2n-2}(C)) inside the nilpotent cone, and its intersection with the stable cotangent chart is the closure of the conormal bundle to the one-spinor stratum of the generalized theta-div
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Graphs of Hecke operators in mixed ramification
Hecke operators on ramified G-bundles in complex ramification mimic simpler cases under mild conditions, reducing the problem to divisors at most two points and allowing dimension computations for PGL_2 eigenforms.
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Gaiotto Loci and the Nilpotent Cone for $\mathrm{Sp}_{2n}(\mathbb C)$
For the standard representation of Sp_{2n}(C), the Gaiotto locus is the Bialynicki-Birula closure associated to U(Sp_{2n-2}(C)) inside the nilpotent cone, and its intersection with the stable cotangent chart is the closure of the conormal bundle to the one-spinor stratum of the generalized theta-div