Derived category D G (X) and functors.- DG-modules and equivariant cohomology.- Equivariant cohomology of toric varieties.

In this article, we present some results and constructions about the Chiu-Tamarkin invariant motivated by the idempotence of microlocal kernels, including: (1) a natural explanation for the definition of the $\mathbb{Z}/\ell$-equivariant Chiu-Tamarkin invariant; (2) a graded commutative product on the non-equivariant Chiu-Tamarkin invariant; and (3) a construction of the $S^1$-equivariant Chiu-Tamarkin invariant. As applications, we: (1) construct a sequence of symplectic capacities $(\overline{c}_k)_{k\in \mathbb{N}}$ and prove that it coincides with the symplectic capacities $({c}_k)_{k\in \mathbb{N}}$ we defined using the $\mathbb{Z}/\ell$-equivariant Chiu-Tamarkin invariant under certain conditions; and (2) prove a Viterbo isomorphism. In the Appendix, we provide a proof of admissibility for all open sets in a cotangent bundle under the setup of triangulated categories.

Equivariant Sheaves and Functors

Idempotence of microlocal kernels and $S^1$-equivariant Chiu-Tamarkin invariant

We consider the principal block of category O and its graded version. On the space of homomorphisms from a Verma module to an indecomposable tilting module we may define natural filtrations following Andersen. The arguments given in this article prove that these filtrations are compatible with the graded structure, although explicitely we only show that the dimensions of the sucessive subquotients of the filtration are compatible with this intuition. This statement is very similar to the semisimplicity of the subquotients of the Jantzen filtration proved by Beilinson and Bernstein, but the method of proof is quite different. I would like to know how to directly relate both results, as this would give an alternative proof of said semisimplicity.

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Andersen-Filtrierung und harter Lefschetz

/pdf/categorical-langlands-correspondence-for-1-2jmyld1ma3.pdf

Categorical Langlands correspondence for _{,1}(ℝ)

The Deligne-Langlands correspondence parametrizes irreducible representations of the affine Hecke algebra $\mathcal{H}^{\text{aff}}$ by certain perverse sheaves. We show that this can be lifted to an equivalence of triangulated categories. More precisely, we construct for each central character $\chi$ of $\mathcal{H}^{\text{aff}}$ an equivalence of triangulated categories between a perfect derived category of dg-modules $D_{\text{perf}}(\mathcal{H}^{\text{aff}}/(\text{ker}(\chi)) - \text{dgMod})$ and the triangulated category generated by the corresponding perverse sheaves. The main step in this construction is a formality result that we prove for a wide range of `Springer sheaves'.

Equivariant Sheaves and Functors

Citations

Idempotence of microlocal kernels and $S^1$-equivariant Chiu-Tamarkin invariant

Andersen-Filtrierung und harter Lefschetz

Categorical Langlands correspondence for _{,1}(ℝ)

Formality in the Deligne-Langlands correspondence

Differential equations and Lie group representations

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