The geometry of nilpotent varieties via subbundles of the cotangent bundle

Abstract: Let $G$ be a simple algebraic group with flag variety $G/B$. The Springer resolution is the moment map from the cotangent bundle of $G/B$ to the (dual of the) Lie algebra $\mathfrak{g}$ of $G$. The cohomology of the fibers of this map play an important role in the representation theory of $G$ over various fields.

Identify the cotangent bundle with the vector bundle $G \times^B \mathfrak{n}$, where $\mathfrak{n}$ is the nilradical of the Lie algebra of $B$. There are subbundles $G \times^B I$ for each subspace $I \subset \mathfrak{n}$ that is $B$-stable, and maps $p_I: G \times^B I \to \mathfrak{g}$. The fibers of $p_I$ are also interesting and their cohomology relates to an intersection cohomology complex on the image of $p_I$, a nilpotent variety.

In this talk we discuss two topics: (1) methods for computing the cohomology of the fibers of $p_I$; (2) a vanishing theorem/conjecture for the cohomology of the structure sheaf on these subbundles.

algebraic geometryalgebraic topologycomplex variablesdifferential geometrygeometric topologymetric geometryquantum algebrarepresentation theory

Audience: researchers in the topic

Sapienza A&G Seminar

Series comments: Weekly research seminar in algebra and geometry.

"Sapienza" Università di Roma, Department of Mathematics "Guido Castelnuovo".