Sporadic discussion

Linear factors for the action of an algebraic group on a split unipotent group

Submitted for publication.

Abstract: Let \(k\) be an arbitrary field, let \(G\) be a linear algebraic group over \(k\), and let \(V\) be a vector group over \(k\) on which \(G\) acts by automorphisms of algebraic groups. The action of \(G\) on \(V\) is said to be linear if there is a \(G\)-equivariant isomorphism of algebraic groups \(V \simeq \operatorname{Lie}(V)\). We give examples of vector groups \(V\) having non-linear action of \(G\). On the other hand, if the \(G\)-module \(A(V)\) of additive regular functions on \(V\) is completely reducible, we show that the action of \(G\) on \(V\) is linear. Using this result, we prove that if the unipotent radical of \(G\) is defined and split over \(k\), then any split unipotent algebraic group \(U\) over \(k\) on which \(G\) acts by group automorphisms has a filtration by \(G\)-stable closed subgroups \(U = U^0 \supset U^1 \supset \cdots\) for which each successive quotient \(U^i/U^{i+1}\) is a vector group having a linear action of \(G\).

• 2008 Fundamental lemma seminar of Hales
• Franz Lemmermeyer has a large compilation of links for lecture notes.
• David Ben-Zvi's page Intro to geometric Langlands
• Paul Smith's page Noncommutative geometry and algebra

Here are the slides from my talk in the special session on algebraic groups (etc) at the Joint Math Meeting in New Orleans, Jan 2011. The talk discussed the following results: if G is a linear algebraic group with "nice enough" unipotent radical, and if G possesses a Levi factor after scalar extension to a separable closure of the ground field, then G already has a Levi factor defined over the ground field.

Here is a graph paper generator available on the web – a handy (?) way to play about with your facets for affine A2 or B2 Weyl groups.

I'm glad that Polo and Gille are (hopefully still) producing a new edition of SGA3.

Here I'm including some links for the PDFs of slides for several recent talks.

• Ascona slides, August 2009 These slides are from my talk at the conference on Algebraic Groups and Invariant Theory in Ascona Switzerland, August 2009.
• Raleigh special session talk from the AMS sectional meeting in Raleigh of April 2009.
• slides for my seminar talk at UMich of Dec 2008.
• slides from my talk in Kalamazoo from the AMS sectional meeting in Kalamazoo MI of Oct 2008.

• Linear Algebraic Groups
• Front for the mathematics arXiv
• JSTOR
• NUMDAM Numérisation de documents anciens mathématiques
• Publications IHES at NUMDAM
• Goettinger DigitalisierungsZentrum (GDZ)