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Zhiwei Yun

Publications and source records attributed to Zhiwei Yun.

At least 55 records · Page 3Linked to original sources
arXiv2016

$\mathbb{Z}/m\mathbb{Z}$-graded Lie algebras and perverse sheaves, III: graded double affine Hecke algebra

In this paper we construct representations of certain graded double affine Hecke algebras (DAHA) with possibly unequal parameters from geometry. More precisely, starting with a simple Lie algebra $\mathfrak{g}$ together with a $\mathbb{Z}/m\mathbb{Z}$-grading $\oplus_{i}\mathfrak{g}_{i}$ and a block of $G_{\underline{0}}$-equivariant complexes on the nilpotent cone of $\mathfrak{g}_{\underline{1}}$ as introduced in \cite{LY1}, we attach a graded DAHA and construct its action on the direct sum of spiral inductions in that block. This generalizes results of Vasserot \cite{V} and Oblomkov-Yun \cite{OY} which correspond to the case of the principal block.

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arXiv2016

Z/m-graded Lie algebras and perverse sheaves, I

We give a block decomposition of the equivariant derived category arising from a cyclically graded Lie algebra. This generalizes certain aspects of the generalized Springer correspondence to the graded setting.

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arXiv2016

$Z/m$-graded Lie algebras and perverse sheaves, II

We consider a fixed block for the equivariant perverse sheaves with nilpotent support in the $1$-graded ccomponent of a semisimple cyclically graded Lie algebra. We give a combinatorial parametrization of the simple objects in that block.

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arXiv2016

Geometric representations of graded and rational Cherednik algebras

We provide geometric constructions of modules over the graded Cherednik algebra $\mathfrak{H}^{gr}_ν$ and the rational Cherednik algebra $\mathfrak{H}^{rat}_ν$ attached to a simple algebraic group $\mathbb{G}$ together with a pinned automorphism $θ$. These modules are realized on the cohomology of affine Springer fibers (of finite type) that admit $\mathbb{C}^*$-actions. In the rational Cherednik algebra case, the standard grading on these modules is derived from the perverse filtration on the cohomology of affine Springer fibers coming from its global analog: Hitchin fibers. When $θ$ is trivial, we show that our construction gives the irreducible finite-dimensional spherical modules $\mathfrak{L}_ν(triv)$ of $\mathfrak{H}^{gr}_ν$ and of $\mathfrak{H}^{rat}_ν$. We give a formula for the dimension of $\mathfrak{L}_ν(triv)$ and give a geometric interpretation of its Frobenius algebra structure. The rank two cases are studied in further details.

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arXiv2016

Lectures on Springer theories and orbital integrals

These are the expanded lecture notes from the author's mini-course during the graduate summer school of the Park City Math Institute in 2015. The main topics covered are: geometry of Springer fibers, affine Springer fibers and Hitchin fibers; representations of (affine) Weyl groups arising from these objects; relation between affine Springer fibers and orbital integrals.

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arXiv2014

On Koszul duality for Kac-Moody groups

For any Kac-Moody group $G$ with Borel $B$, we give a monoidal equivalence between the derived category of $B$-equivariant mixed complexes on the flag variety $G/B$ and (a certain completion of) the derived category of $B^\vee$-monodromic mixed complexes on the enhanced flag variety $G^\vee/U^\vee$, here $G^\vee$ is the Langlands dual of $G$. We also prove variants of this equivalence, one of which is the equivalence between the derived category of $U$-equivariant mixed complexes on the partial flag variety $G/P$ and certain "Whittaker model" category of mixed complexes on $G^\vee/B^\vee$. In all these equivalences, intersection cohomology sheaves correspond to (free-monodromic) tilting sheaves. Our results generalize the Koszul duality patterns for reductive groups in the work of Beilinson, Ginzburg and Soergel .

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arXiv2014

Rigidity in automorphic representations and local systems

We introduce the notion of rigidity for automorphic representations of groups over global function fields. We construct the Langlands parameters of rigid automorphic representations explicitly as local systems over open curves. We expect these local systems to be rigid. Examples of rigid automorphic representations from previous work are reviewed and more examples for $GL_{2}$ are discussed in details.

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arXiv2014

Epipelagic representations and rigid local systems

We construct automorphic representations for quasi-split groups $G$ over the function field $F=k(t)$ one of whose local components is an epipelagic representation in the sense of Reeder and Yu. We also construct the attached Galois representations under the Langlands correspondence. These Galois representations give new classes of conjecturally rigid, wildly ramified $^{L}G$-local systems over $\mathbb{P}^{1}-\{0,\infty\}$ that generalize the Kloosterman sheaves constructed earlier by Heinloth, Ngô and the author. We study the monodromy of these local systems and compute all examples when $G$ is a classical group.

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arXiv2013

Galois representations attached to moments of Kloosterman sums and conjectures of Evans

Kloosterman sums for a finite field arise as Frobenius trace functions of certain local systems defined over $\Gm$. The moments of Kloosterman sums calculate the Frobenius traces on the cohomology of tensor powers (or symmetric powers, exterior powers, etc.) of these local systems. We show that when $p$ ranges over all primes, the moments of the corresponding Kloosterman sums for $\mathbb{F}_{p}$ arise as Frobenius traces on a continuous $\ell$-adic Galois representation that comes from geometry. We also give bounds on the ramification of these Galois representations. All this is done in the generality of Kloosterman sheaves attached to reductive groups introduced in \cite{HNY}. As an application, we give proofs of conjectures of R. Evans (\cite{Evans0}, \cite{Evans}) expressing the seventh and eighth symmetric power moments of the classical Kloosterman sum in terms of Fourier coefficients of explicit modular forms. The proof for the eighth symmetric power moment conjecture relies on the computation done in Appendix B by C.Vincent.

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arXiv2013

Orbital integrals and Dedekind zeta functions

Let F be a local non-archimedean field. We prove a formula relating orbital integrals in GL(n,F) (for the unit Hecke function) and the generating series counting ideals of a certain ring. Using this formula, we give an explicit estimate for such orbital integrals. We also derive an analogous formula for global fields, proving analytic properties of the Dedekind zeta function for orders in global fields.

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arXiv2012

A $(-q)$-analogue of weight multiplicities

We prove a conjecture in \cite{L} stating that certain polynomials $P^σ_{y,w}(q)$ introduced in \cite{LV1} for twisted involutions in an affine Weyl group give $(-q)$-analogues of weight multiplicities of the Langlands dual group $\check{G}$. We also prove that the signature of a naturally defined hermitian form on each irreducible representation of $\check{G}$ can be expressed in terms of these polynomials $P^σ_{y,w}(q)$.

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arXiv2012

Fake degrees for reflection actions on roots

A finite irreducible real reflection group of rank l and Coxeter number h has root system of cardinality h*l. It is shown that the fake degree for the permutation action on its roots is divisible by [h]_q = 1+q+q^2+...+q^{h-1}, and that in simply-laced types, it equals [h]_q times the summation of q^{e_i - 1} where e_i runs through the exponents, so that e_i - 1 are the codegrees.

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arXiv2011

Motives with exceptional Galois groups and the inverse Galois problem

We construct motivic $\ell$-adic representations of $\GQ$ into exceptional groups of type $E_7,E_8$ and $G_2$ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and uses the Langlands correspondence for function fields. As an application, we solve new cases of the inverse Galois problem: the finite simple groups $E_{8}(\FF_{\ell})$ are Galois groups over $\QQ$ for large enough primes $\ell$.

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arXiv2011

Macdonald formula for curves with planar singularities

We generalize Macdonald's formula for the cohomology of Hilbert schemes of points on a curve from smooth curves to curves with planar singularities: we relate the cohomology of the Hilbert schemes to the cohomology of the compactified Jacobian of the curve. The new formula is a consequence of a stronger identity between certain perverse sheaves defined by a family of curves satisfying mild conditions. The proof makes an essential use of Ngo's support theorem for compactified Jacobians and generalizes this theorem to the relative Hilbert scheme of such families. As a consequence, we generalize part of the Weil conjectures to the Hilbert-zeta function of curves with planar singularities.

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arXiv2011

The spherical part of the local and global Springer actions

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 that in both situations, the action of the center of the group algebra of the affine Weyl group (the spherical part) factors through the action of the component group of the relevant centralizers. In the situation of affine Springer fibers, this partially verifies a conjecture of Goresky-Kottwitz-MacPherson and Bezrukavnikov-Varshavsky. We first prove this result for the global Springer action, and then deduce the result for the local Springer action from that of the global one. The argument strongly relies on the fact that we can deform points on a curve, hence giving an example of using global Springer theory to solve more classical problems.

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arXiv2010

Kloosterman sheaves for reductive groups

Deligne constructed a remarkable local system on $\bP^1-\{0,\infty\}$ attached to a family of Kloosterman sums. Katz calculated its monodromy and asked whether there are Kloosterman sheaves for general reductive groups and which automorphic forms should be attached to these local systems under the Langlands correspondence. Motivated by work of Gross and Frenkel-Gross we find an explicit family of such automorphic forms and even a simple family of automorphic sheaves in the framework of the geometric Langlands program. We use these automorphic sheaves to construct l-adic Kloosterman sheaves for any reductive group in a uniform way, and describe the local and global monodromy of these Kloosterman sheaves. In particular, they give motivic Galois representations with exceptional monodromy groups G_2,F_4,E_7 and E_8. This also gives an example of the geometric Langlands correspondence with wild ramifications for any reductive group.

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arXiv2009

Integral homology of loop groups via Langlands dual groups

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.

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