Tag - Representations of algebraic groups

Scott Larson: Small Resolutions of Closures of K-Orbits in Flag Varieties I

14th September, 2020

The geometry of closures of K-orbits in the flag variety governs key properties in representation theory of real reductive groups. For example, Kazhdan-Lusztig-Vogan polynomials and characteristic cycles of Harish-Chandra modules are of current interest but difficult to compute. Barbasch-Evens constructed resolutions for K-orbits on grassmannian flag varieties and found some small resolutions. We do the same thing for isotropic flag varieties of the symplectic group, where K=GLn. This leads us to describe natural resolutions for K-orbits, generalizing many constructions found in the literature.

Antoine Touzé: Stabilization and cup products for polynomial representations of GLn(k)

15th August, 2016

It is known for a long time that polynomial representations of GLn(k) stabilize when n grows, i.e. Schur algebras S(n, d) are all Morita equivalent when n ≥ d. A model of the category of stable polynomial representations is given by the strict polynomial functors of Friedlander and Suslin. Using the formalism of strict polynomial functors, we prove a rather counter-intuitive results on cup products, namely that the cup product

Ext∗(M, N) ⊗ Ext∗(P(r), Q(r)) → Ext∗(M ⊗ P(r), N ⊗ Q(r))

induces an isomorphism in low degrees when M, N, P, Q are stable polynomial representations. We shall explain some consequences of these results (including a new proof of the Steinberg tensor product theorem, as well as more general structure theorems which generalize it) and connections with the cohomology of the symmetric group.

Michel Brion: The isogeny category of commutative algebraic groups

10th August, 2016

The commutative algebraic groups over a prescribed field k form an abelian category Ck; the finite commutative algebraic groups form a full  subcategory Fk, stable under taking subobjects, quotients and  extensions. This mini-course will study the categories Ck and  Ck/Fk (the isogeny category) from a homological viewpoint, emphasizing the analogies and differences with  categories of representations. In particular, we will show that Ck/Fk has homological dimension 1, and we will describe the projective and the injective objects in Ck and Ck/Fk.