My colleague David Stewart organized a workshop on Pseudo-reductive groups in September 2017, which was partially funded by the Heilbronn Institute.
In this workshop, Gopal Prasad gave a mini-course on his work with Conrad and Gabber on pseudo-reductive groups.
I contributed a lecture on Reductive subgroups of parahoric group schemes. Here is the abstract for my talk:
Let \(K\) be the field of fractions of a complete discrete valuation ring \(A\) with residue field \(k\), and let \(G\) be a connected reductive algebraic group over \(K\). Suppose \(P\) is a parahoric group scheme attached to \(G\). In particular, \(P\) is a smooth affine \(A\)-group scheme having generic fiber \(P_K = G\); the group scheme \(P\) is in general not reductive over \(A\). Assume that \(G\) splits over an unramified extension of \(K\).
The talk will give an overview of two results about \(G\).
First, there is a closed and reductive \(A\)-subgroup scheme \(M\) of \(P\) for which the special fiber \(M_k\) is a Levi factor of \(P_k\). Moreover, the reductive subgroups of \(G=P_K\) of the form \(M_K\) may be characterized.
Second, let \(X\) be a nilpotent section in \(\operatorname{Lie}(P)\). We say that \(X\) is balanced if the fibers \(C_K\) and \(C_k\) are smooth group schemes of the same dimension, where \(C=C_P(X)\) is the scheme theoretic centralizer of \(X\). If \(X_0\) is a given nilpotent element in the Lie algebra of the reductive quotient of the special fiber \(P_k\), we give results on the possible lifts of \(X_0\) to a balanced nilpotent section \(X \in \operatorname{Lie}(P)\).