arXiv subjects
Publications and source records attributed to Xinwen Zhu.
We show that the irregular connection on $\bbG_m$ constructed by Frenkel-Gross (2009) and the one constructed by Heinloth-Ngô-Yun (arXiv:1005.2765) are the same, which confirms the Conjecture 2.14 of arXiv:1005.2765.
We define and study the 2-category of torsors over a Picard groupoid, a central extension of a group by a Picard groupoid, and commutator maps in this central extension. Using it in the context of two-dimensional local fields and two-dimensional adelic theory we obtain the two-dimensional tame symbol and a new proof of Parshin reciprocity laws on an algebraic surface.
We prove that any flat G-bundle, where G is a complex connected reductive algebraic group, on the punctured disc admits the structure of an oper. This result is important in the local geometric Langlands correspondence proposed in arXiv:math/0508382. Our proof uses certain deformations of the affine Springer fibers which could be of independent interest. As a byproduct, we construct representations of affine Weyl groups on the homology of these deformations generalizing representations constructed by Lusztig.
For any abelian category \calC satsifying (AB5) over a separated, quasi-compact scheme S, we construct a stack of 2-groups \GL(\calC) over the flat site of S. We will give a concrete description of \GL(\calC) when \calC is the category of quasi-coherent sheaves on a separated, quasi-compact scheme X over S. We will show that the tangent space \gl(\calC) of \GL(\calC) at the origin has a structure as a Lie 2-algebra.
Let G^\vee be a complex simple algebraic group. We describe certain morphisms of G^\vee(\calO)-equivariant complexes of sheaves on the affine Grassmannian \Gr of G^\vee in terms of certain morphisms of G-equivariant coherent sheaves on \frakg, where G is the Langlands dual group of G^\vee and \frakg is its Lie algebra. This can be regarded as an example of the derived Satake correspondence.
Let K be a connected compact Lie group, and G be its complexification. The homology of the based loop group ΩK with integer coefficients is naturally a \ZZ-Hopf algebra. After possibly inverting 2 or 3, we identify H_*(ΩK,\ZZ) with the Hopf algebra of algebraic functions on B^\vee_e, where B^\vee is a Borel subgroup of the Langlands dual group scheme G^\vee of G and B^\vee_e is the centralizer in B^\vee of a regular nilpotent element e\in\Lie B^\vee. We also give a similar interpretation for the equivariant homology of ΩK under the maximal torus action.
Let $G$ be a simple algebraic group of type $A$ or $D$ defined over $\C$ and $T$ be a maximal torus of $G$. For a dominant coweight $λ$ of $G$, the $T$-fixed point subscheme $(\bar{Gr}_G^λ)^T$ of the Schubert variety $\bar{Gr}_G^λ$ in the affine Grassmannian $Gr_G$ is a finite scheme. We prove that there is a natural isomorphism between the dual of the level one affine Demazure module corresponding to $λ$ and the ring of functions (twisted by certain line bundle on $Gr_G$) of $(\bar{Gr}_G^λ)^T$. We use this fact to give a geometric proof of the Frenkel-Kac-Segal isomorphism between basic representations of affine algebras of $A,D,E$ type and lattice vertex algebras.
A crucial role in representation theory of loop groups of reductive Lie groups and their Lie algebras is played by their non-trivial second cohomology classes which give rise to their central extensions (the affine Kac-Moody groups and Lie algebras). Loop groups embed into the group GL_\infty of continuous automorphisms of C((t)), and these classes come from a second cohomology class of GL_\infty. In a similar way, double loop groups embed into a group of automorphisms of C((t))((s)), denoted by GL_{\infty,\infty}, which has a non-trivial third cohomology. In this paper we explain how to realize a third cohomology class in representation theory of a group: it naturally arises when we consider representations on categories rather than vector spaces. We call them "gerbal representations." We then construct a gerbal representation of GL_{\infty,\infty} (and hence of double loop groups), realizing its non-trivial third cohomology class, on a category of modules over an infinite-dimensional Clifford algebra. This is a two-dimensional analogue of the fermionic Fock representations of the ordinary loop groups.