MATH Seminar

Title: The special fiber of a parahoric group scheme
Seminar: Algebra and number theory
Speaker: George McNinch of Tufts University
Contact: Skip Garibaldi, skip@mathcs.emory.edu
Date: 2010-10-26 at 3:00PM
Venue: MSC E408
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Abstract: Let $G$ be a connected and reductive algebraic group over the field of fractions $K$ of a complete discrete valuation ring $A$ with residue field $k$. Bruhat and Tits have associated with $G$ certain smooth $A$-group schemes $P$ --- called parahoric group schemes --- which have generic fiber $P/K = G$. The special fiber $P/k$ of such a group scheme is a linear algebraic group over $k$, and in general it is not reductive. In some recent work, it was proved that $P/k$ has a Levi factor in case $G$ splits over an unramified extension of $K$. Even more recently, this result was (partially) extended to cover the case where G splits over a tamely ramified extension. The talk will discuss these results and some applications. In particular, it will mention possible applications to the description of the scheme-theoretic centralizer of suitable nilpotent sections in Lie$(P)(A)$.