Tag - Geometric representation theory

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

21st 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. We recall how small resolutions have been used to compute these invariants, describe fibres of the resolutions from last week, and describe more small resolutions for the real reductive groups Sp2n(ℝ), U(p,q), and complex groups. Along the way we consider an application to Schubert varieties.

Chris Bowman: Tautological p-Kazhdan-Lusztig Theory for cyclotomic Hecke algebras

15th September, 2020

We discuss a new explicit isomorphism between (truncations of) quiver Hecke algebras and Elias-Williamson’s diagrammatic endomorphism algebras of Bott-Samelson bimodules. This allows us to deduce that the decomposition numbers of these algebras (including as examples the symmetric groups and generalised blob algebras) are tautologically equal to the associated p-Kazhdan-Lusztig polynomials, provided that the characteristic is greater than the Coxeter number. This allows us to give an elementary and explicit proof of the main theorem of Riche-Williamson’s recent monograph and extend their categorical equivalence to cyclotomic Hecke algebras, thus solving Libedinsky-Plaza’s categorical blob conjecture.

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.

Frances Kirwan: Variation of non-reductive geometric invariant theory

29th April, 2017

Mumford's geometric invariant theory (GIT) provides a method for constructing quotient varieties for linear actions of reductive groups on projective varieties. The GIT quotient depends on the choice of linearization for the group action, and this dependence was described using 'variation of GIT' (VGIT) by Thaddeus and Dolgachev & Hu in the 1990s. GIT has been extended to non-reductive actions; many of the nice features of classical GIT fail in general, but are satisfied given the extra data of a graded linearization for an action of a linear algebraic group with graded unipotent radical. The aim of this talk is to describe this picture and a version of VGIT which applies to it.

Andrei Okounkov: Weyl groups and their generalizations in enumerative geometry

15th March, 2016

These lectures will be about enumerative K-theory of curves (and more general 1-dimensional sheaves) in algebraic threefolds. In the first lecture, we will set up the enumerative problem and survey what we know and what we conjecture about it. In particular, we will meet the fundamental building blocks of the theory: threefolds fibered in ADE surfaces. In the second lecture, we will learn what geometric representation theory says about these building blocks, and, in particular, meet the present day incarnation of the Weyl group, which is really a fundamental groupoid of a certain periodic hyperplane arrangement, associated to a certain geometrically defined infinite-dimensional Lie algebra. This Weyl group completely determines the curve counts, and so seems like a very fitting topic for Hermann Weyl lectures. In the third lecture, I plan to introduce some of the geometric ideas that go into the actual technical construction of the theory.

Jacob Lurie: Categorification of the Fourier Transform

24th April, 2015

In this talk, I will discuss the process of 'categorification'; that is, taking a known statement about concrete objects (like sets) and looking for a generalization (or analogue) in a more abstract setting. I will give some specific examples, beginning with the classical theory of the Fourier transform, and (if time permits) briefly discuss how the 'categorification' of this theory arises in geometric representation theory.

Jacob Lurie: Categorification of the Fourier Transform

24th February, 2015

In this talk, I will discuss the process of 'categorification'; that is, taking a known statement about concrete objects (like sets) and looking for a generalization (or analogue) in a more abstract setting. I will give some specific examples, beginning with the classical theory of the Fourier transform, and (if time permits) briefly discuss how the 'categorification' of this theory arises in geometric representation theory.