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

Kbe the field of fractions of a complete discrete valuation ringAwith residue fieldk, and letGbe a connected reductive algebraic group overK. SupposePis a parahoric group scheme attached toG. In particular,Pis a smooth affineA-group scheme having generic fiberP_{K}=G; the group schemePis in general not reductive overA. Assume thatGsplits over an unramified extension ofK.

The talk will give an overview of two results about

G.

First, there is a closed and reductive

A-subgroup schemeMofPfor which the special fiberM_{k}is a Levi factor ofP_{k}. Moreover, the reductive subgroups ofG=P_{K}of the formM_{K}may be characterized.

Second, let

Xbe a nilpotent section in Lie (P). We say thatXis balanced if the fibersC_{K}andC_{k}are smooth group schemes of the same dimension, whereC=C_{P}(X) is the scheme theoretic centralizer ofX. IfX_{0}is a given nilpotent element in the Lie algebra of the reductive quotient of the special fiberP_{k}, we give results on the possibleliftsofX_{0}to a balanced nilpotent sectionX∈ Lie (P).