Cycle relations in the affine grassmannian and applications to p-adic Galois representations

Robin Bartlett (University of Glasgow)

Wednesday 1st November, 2023 16:00-17:00 Maths 110


In the first half of the talk I will introduce the affine grassmannian and explain a result which describes multiplicities in e-fold tensor products of Weyl modules in terms of cycle relations between certain closed subschemes inside the affine grassmannian. The closed subschemes are constructed as degenerations of e-fold products of flag varieties, and in the second half of the talk I will motivate their appearance by showing how, when the base field has characteristic p, their geometry models that of certain moduli spaces of p-adic representations of the Galois group of a p-adic field K/Qp. As an application we use our original result to deduce new cases of a conjecture, the Breuil—M'ezard conjecture, which asserts that congruences between automorphic forms are mirrored by congruences between p-adic Galois representations. This can be viewed as a combinatorial shadow of the expected p-adic Langlands correspondence. 

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