I give two talks at the Northern Illinois University mathematics colloquium.

Here are the abstracts, and links for the slides (as pdf documents):

Group cohomology and Levi decompositions for linear groups (Mar 21)

*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.Some tools for the study of reductive groups over local fields (Mar 22)

*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*P*_{k}over*k*. When*P*is not reductive, we investigate the question: does*P*_{k}have a Levi decomposition (as in the first talk)? This second talk will include a good bit of example-oriented background discussion.