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Zhiwei Yun

Publications and source records attributed to Zhiwei Yun.

At least 37 records · Page 2Linked to original sources
arXiv2023

Higher Siegel--Weil formula for unitary groups: the non-singular terms

We construct special cycles on the moduli stack of unitary shtukas. We prove an identity between (1) the r-th central derivative of non-singular Fourier coefficients of a normalized Siegel--Eisenstein series, and (2) the degree of special cycles of "virtual dimension 0" on the moduli stack of unitary shtukas with r legs. This may be viewed as a function-field analogue of the Kudla-Rapoport Conjecture, that has the additional feature of encompassing all higher derivatives of the Eisenstein series.

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arXiv2023

Tilting sheaves for real groups and Koszul duality

For a certain class of real analytic varieties with Lie group actions we develop a theory of (free-monodromic) tilting sheaves, and apply it to flag varieties stratified by real group orbits. For quasi-split real groups, we construct a fully faithful embedding of the category of tilting sheaves to a real analog of the category of Soergel bimodules, establishing real group analogs of Soergel's Structure Theorem and Endomorphism Theorem. We apply these results to give a purely geometric proof of the theorem of Bezrukavnikov and Vilonen which proves Soergel's conjecture for quasi-split groups.

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arXiv2023

Automorphic gluing functor in Betti Geometric Langlands

We study automorphic categories of nilpotent sheaves under degenerations of smooth curves to nodal Deligne-Mumford curves. Our constructions realize affine Hecke operators as the result of bubbling projective lines from marked points. We use this to construct a "gluing functor" from the automorphic category of a nodal Deligne-Mumford curve to the automorphic category of a smoothing.

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arXiv2023

Euphotic representations and rigid automorphic data

We propose a new method to construct rigid $G$-automorphic representations and rigid $\widehat{G}$-local systems for reductive groups $G$. The construction involves the notion of euphotic representations, and the proof for rigidity involves the geometry of certain Hessenberg varieties.

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arXiv2022

Non-abelian Hodge moduli spaces and homogeneous affine Springer fibers

Starting from a homogeneous affine Springer fiber $Fl_ψ$, we construct three moduli spaces that correspond to the Dolbeault, de Rham and Betti aspects of a hypothetical Simpson correspondence with wild ramifications. We show that $Fl_ψ$ is homeomorphic to the central Lagrangian fiber in the Dolbeault space, prove that the Dolbeaut and de Rham spaces both have the same cohomology as $Fl_ψ$, and construct a map from the de Rham space to the Betti space which we conjecture to be an analytic isomorphism.

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arXiv2022

Rigidity method for automorphic forms over function fields

These are lectures given at the 2022 Arizona Winter School. It gives an introduction to the rigidity method for constructing automorphic forms for semisimple groups over function fields. The rigidity method leads to explicit constructions of local systems that are Langlands parameters of automorphic forms. The examples of local systems produced in this way have applications to algebraic geometry and number theory.

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arXiv2022

Special cycles for Shtukas are closed

In this paper we give a different proof of a theorem of Paul Breutmann: for a Bruhat-Tits group scheme $\mathcal{H}$ over a smooth projective curve $X$ and a closed embedding into another smooth affine group scheme $\mathcal{G}$, the induced map on the moduli of Shtukas $Sht^{r}_{\mathcal{H}}\to Sht^{r}_{\mathcal{G}}$ is schematic, finite and unramified. This result enables one to define special cycles on the moduli stack of Shtukas.

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arXiv2021

Endoscopy for Hecke categories, character sheaves and representations

For a split reductive group $G$ over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group $H$ with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on $G$ with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group $H$, after passing to asymptotic versions. The other is a similar relationship between representations of $G(\mathbb{F}_q)$ with a fixed semisimple parameter and unipotent representations of $H(\mathbb{F}_{q})$.

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arXiv2020

Shtukas and the Taylor expansion of $L$-functions (II)

For arithmetic applications, we extend and refine our results in \cite{YZ} to allow ramifications in a minimal way. Starting with a possibly ramified quadratic extension $F'/F$ of function fields over a finite field in odd characteristic, and a finite set of places $Σ$ of $F$ that are unramified in $F'$, we define a collection of Heegner--Drinfeld cycles on the moduli stack of $\mathrm{PGL}_{2}$-Shtukas with $r$-modifications and Iwahori level structures at places of $Σ$. For a cuspidal automorphic representation $π$ of $\mathrm{PGL}_{2}(\mathbb{A}_{F})$ with square-free level $Σ$, and $r\in\mathbb{Z}_{\ge0}$ whose parity matches the root number of $π_{F'}$, we prove a series of identities between: (1) The product of the central derivatives of the normalized $L$-functions $\mathcal{L}^{(a)}(π, 1/2)\mathcal{L}^{(r-a)}(π\otimesη, 1/2)$, where $η$ is the quadratic idèle class character attached to $F'/F$, and $0\le a\le r$; (2) The self intersection number of a linear combination of Heegner--Drinfeld cycles. In particular, we can now obtain global $L$-functions with odd vanishing orders. These identities are function-field analogues of the formulas of Waldspurger and Gross--Zagier for higher derivatives of $L$-functions.

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arXiv2019

Semilinear automorphisms of classical groups and quivers

For a classical group $G$ over a field $F$ together with a finite-order automorphism $θ$ that acts compatibly on $F$, we describe the fixed point subgroup of $θ$ on $G$ and the eigenspaces of $θ$ on the Lie algebra $\mathfrak{g}$ in terms of cyclic quivers with involution. More precise classification is given when $\mathfrak{g}$ is a loop Lie algebra, i.e., when $F=\mathbb{C}((t))$.

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arXiv2019

Higher signs for Coxeter groups

We define and study cocycles on a Coxeter group in each degree generalizing the sign function. When the Coxeter group is a Weyl group, we explain how the degree three cocycle arises naturally from geometry representation theory.

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arXiv2019

Spectral action in Betti Geometric Langlands

Let $X$ be a smooth projective curve, $G$ a reductive group, and $Bun_G(X)$ the moduli of $G$-bundles on $X$. For each point of $X$, the Satake category acts by Hecke modifications on sheaves on $Bun_G(X)$. We show that, for sheaves with nilpotent singular support, the action is locally constant with respect to the point of $X$. This equips sheaves with nilpotent singular support with a module structure over perfect complexes on the Betti moduli $Loc_{G^\vee}(X)$ of dual group local systems. In particular, we establish the "automorphic to Galois" direction in the Betti Geometric Langlands correspondence -- to each indecomposable automorphic sheaf, we attach a dual group local system -- and define the Betti version of V. Lafforgue's excursion operators.

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arXiv2018

Hitchin type moduli stacks in automorphic representation theory

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 on applications of this observation to a relative fundamental lemma, the arithmetic fundamental lemma and to the higher Gross-Zagier formula.

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arXiv2018

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

Let G be a reductive group over C. Assume that the Lie algebra g of G has a given grading (g_j) indexed by a cyclic group Z/m such that g_0 contains a Cartan subalgebra of g. The subgroup G_0 of G corresponding to g_0 acts on the variety of nilpotent elements in g_1 with finitely many orbits. We are interested in computing the local intersection cohomology of the closures of these orbits with coefficients in irreducible G-equivariant local systems which are in the "principal block". We show that these can be computed by a purely combinatorial algorithm.

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arXiv2017

The cohomology ring of certain compactified Jacobians

We provide an explicit presentation of the equivariant cohomology ring of the compactified Jacobian $J_{q/p}$ of the rational curve $C_{q/p}$ with planar equation $x^{q}=y^{p}$ for $(p,q)=1$. We also prove analogous results for the closely related affine Springer fiber $Sp_{q/p}$ in the affine flag variety of $SL_{p}$. We show that the perverse filtration on the cohomology of $J_{q/p}$ is multiplicative, and the associated graded ring under the perverse filtration is a degeneration of the ring of functions on a moduli space of maps $\mathbb{P}^{1}\to C_{q/p}$. We also propose several conjectures about $J_{q/p}$ and more general compactified Jacobians.

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arXiv2017

Shtukas and the Taylor expansion of $L$-functions

We define the Heegner--Drinfeld cycle on the moduli stack of Drinfeld Shtukas of rank two with $r$-modifications for an even integer $r$. We prove an identity between (1) The $r$-th central derivative of the quadratic base change $L$-function associated to an everywhere unramified cuspidal automorphic representation $π$ of $PGL_{2}$; (2) The self-intersection number of the $π$-isotypic component of the Heegner--Drinfeld cycle. This identity can be viewed as a function-field analog of the Waldspurger and Gross--Zagier formula for higher derivatives of $L$-functions.

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