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Reductive subgroup schemes of a parahoric group scheme

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George McNinch
Tufts University
Sat, 2018-11-24 10:30 - 11:30
MPIM Lecture Hall
Let K be the field of fractions of a complete discrete valuation ring A with residue field k, and let G be a connected and reductive linear algebraic group over K. Bruhat-Tits associate to G various parahoric group schemes P. Such P are smooth and affine group schemes over A, but in general they are not reductive.

Assume that G splits over an unramified extension of K and that P is one of these parahoric group schemes. Under these assumptions, we prove that there is a reductive subgroup scheme M of P such that M_k is a Levi
factor of the special fiber P_k, and such that M_K is a reductive subgroup of G containing a maximal torus. In fact, M_K is - at least geometrically - the centralizer of the image of a homomorphism μ_N → G
for some N > 1.

The talk will describe the construction of M, and it will describe some application of the existence of M to the study of G(K)-orbits on nilpotent elements of Lie(G).
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