I give two talks at the Northern Illinois University mathematics colloquium.
Here are the abstracts, and links for the slides (as pdf documents):
Abstract: If G is a linear algebraic group over a field F, we describe what the Hochschild cohomology of G with coefficients in linear representations of G says about those algebraic groups which are extensions of G by connected unipotent algebraic groups over F. If G is reductive and if F has characteristic zero – say, if F is the field of complex numbers – one knows that every such extension is trivial. But if F has positive characteristic, there are non-trivial extensions – i.e. there are linear groups with no Levi decomposition. The talk will give details and examples about these notions and results.
Abstract: Let K be a local field – i.e. the field of fractions of a complete discrete valuation ring A. The study of linear algebraic groups G over such fields K has applications in number theory and algebraic geometry. Some reductive groups (“split groups”) have models over A which are reductive. But e.g. if G does not become split upon base change with any unramified extension of K, it can happen that G has no reductive model. Our interest here is in an interesting family of models for G – the so-called parahoric group schemes P. If k denotes the residue field of A, then by “base-change”, P determines a linear algebraic group Pk over k. When P is not reductive, we investigate the question: does Pk have a Levi decomposition (as in the first talk)? This second talk will include a good bit of example-oriented background discussion.