Algebraic and Symplectic Geometry Seminar

The bounded coherent dg-category on (suitable versions of) the Steinberg stack of a reductive group G is a categorification of the affine Hecke algebra in representation theory.  We discuss how to describe the center and universal trace of this monoidal dg-category.  Many of the techniques involved are very general, and the description makes use of the notion of "odd micro-support" of coherent complexes.  This is joint work with Ben-Zvi and Nadler.

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.

In the third talk we will see that $\Lambda$ admits a canonical quantization if it is a "conification" of a compact exact Lagrangian submanifold of a

cotangent bundle. We will see how to use this quantization to recover results of Fukaya-Seidel-Smith and Abouzaid on the topology of $\Lambda$.

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.

In the second talk we will introduce a stack on $\Lambda$ by localization of the category of sheaves on $M$. We deduce topological obstructions on $\Lambda$ for the existence of a quantization.