# Global Springer theory

@article{Yun2011GlobalST, title={Global Springer theory}, author={Zhiwei Yun}, journal={Advances in Mathematics}, year={2011}, volume={228}, pages={266-328} }

Abstract We generalize Springer representations to the context of groups over a global function field. The global counterpart of the Grothendieck simultaneous resolution is the parabolic Hitchin fibration. We construct an action of the graded double affine Hecke algebra (DAHA) on the direct image complex of the parabolic Hitchin fibration. In particular, we get representations of the degenerate graded DAHA on the cohomology of parabolic Hitchin fibers, providing the first step towards a global… Expand

#### 32 Citations

The spherical part of the local and global Springer actions

- Mathematics
- 2011

The affine Weyl group acts on the cohomology (with compact support) of affine Springer fibers (local Springer theory) and of parabolic Hitchin fibers (global Springer theory). In this paper, we show… Expand

Langlands duality and global Springer theory

- Mathematics
- Compositio Mathematica
- 2012

Abstract We compare the cohomology of (parabolic) Hitchin fibers for Langlands dual groups G and G∨. The comparison theorem fits in the framework of the global Springer theory developed by the… Expand

A global analogue of the Springer resolution for SL2

- Mathematics
- 2012

The global nilpotent cone N is a singular stack associated to the choice of an algebraic group G, a smooth projective curve X, and a line bundle L on X, which is of fundamental importance to the… Expand

Ginzburg algebras and the Hitchin connection for parabolic G-bundles

- Mathematics, Physics
- 2021

For a simple, simply connected, complex group G, we prove the existence of a flat projective connection on the bundle of nonabelian theta functions on the moduli space of semistable parabolic… Expand

Formality for the nilpotent cone and a derived Springer correspondence

- Mathematics
- 2012

Abstract Recall that the Springer correspondence relates representations of the Weyl group to perverse sheaves on the nilpotent cone. We explain how to extend this to an equivalence between the… Expand

Hilbert schemes on plane curve singularities are generalized affine Springer fibers

- Mathematics, Physics
- 2020

In this paper, we show that Hilbert schemes of planar curve singularities can be interpreted as generalized affine Springer fibers for $GL_n$. This leads to a construction of a rational Cherednik… Expand

Affine Springer Fibers, Procesi bundles, and Cherednik algebras

- Mathematics
- 2021

Let g be a semisimple Lie algebra, t its Cartan subalgebra and W the Weyl group. The goal of this paper is to prove an isomorphism between suitable completions of the equivariant Borel-Moore homology… Expand

Riemann–Hilbert for tame complex parahoric connections

- Mathematics
- 2010

A local Riemann–Hilbert correspondence for tame meromorphic connections on a curve compatible with a parahoric level structure will be established. Special cases include logarithmic connections on… Expand

Hitchin type moduli stacks in automorphic representation theory

- Mathematics
- 2018

In the study of automorphic representations over a function field, Hitchin moduli stack and its variants naturally appear and their geometry helps the comparison of trace formulae. We give a survey… Expand

On the construction of moduli stack of projective Higgs bundles over surfaces.

- Mathematics
- 2019

We generalize the construction of M. Lieblich for the compactification of the moduli stack of $\PGL_r$-bundles on algebraic spaces to the moduli stack of Tanaka-Thomas $\PGL_r$-Higgs bundles on… Expand

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We propose a generalization of Springer representations to the context of groups over a global function field. The global counterpart of the Grothendieck simultaneous resolution is the parabolic… Expand

Towards a Global Springer Theory II: the double affine action

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We construct an action of the graded double affine Hecke algebra (DAHA) on the parabolic Hitchin complex, extending the affine Weyl group action constructed in \cite{GSI}. In particular, we get… Expand

Langlands duality and global Springer theory

- Mathematics
- Compositio Mathematica
- 2012

Abstract We compare the cohomology of (parabolic) Hitchin fibers for Langlands dual groups G and G∨. The comparison theorem fits in the framework of the global Springer theory developed by the… Expand

Towards a Global Springer Theory III: Endoscopy and Langlands duality

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