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Taiwang Deng

Publications and source records attributed to Taiwang Deng.

At least 19 recordsLinked to original sources
arXiv2026

Euler Characteristics of $\mathrm{SL}_4(\mathbb{Z})$ and $\mathrm{GL}_4(\mathbb{Z})$, and their cohomological consequences

We compute the homological Euler characteristics of $\mathrm{SL}_4(\mathbb{Z})$ and $\mathrm{GL}_4(\mathbb{Z})$ with coefficients in arbitrary irreducible rational highest-weight representations. Applying Wall's formula, we combine the orbifold Euler characteristics of centralizers of torsion elements with traces computed using the Jacobi-Trudi identity to derive explicit formulas and rational generating functions. Consequently, these Euler characteristics are quasi-polynomial functions of the highest-weight parameters, of total degree at most two. We further derive degreewise vanishing results and parity-sensitive lower bounds for the dimensions of cohomology groups. The two extensions of an $\mathrm{SL}_4(\mathbb{Z})$-coefficient system to $\mathrm{GL}_4(\mathbb{Z})$ yield sharper bounds, which grow linearly or quadratically in explicit infinite families. For symmetric powers, combining our formulas with Horozov's calculation of the determinant-twisted summand yields exact identities and lower bounds for the untwisted summand, together with a conjectural degreewise description of its cohomology.

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arXiv2026

Yun's zeta function and the overorder zeta function for Gorenstein cubic orders

For every Gorenstein cubic $\mathbb Z$-order, we prove an identity equating Yun's zeta function, defined by counting finite-index submodules of the trace dual, with the explicit overorder zeta function introduced in our previous work on Beyond Endoscopy for $\mathrm{GL}_3$. This is posed as Conjecture A in an early draft of Deng-Espinosa and Lee subsequently proved the functional equation of the overorder zeta function, making the Deng--Espinosa isolation of the trivial representation fully unconditional. His argument computes the local factors explicitly. Our proof is independent of Lee's and does not evaluate the individual cubic overorder factors: it matches natural decompositions of the two sides and concludes by induction. As an application, we give a short, uniform evaluation of the local $\mathrm{GL}_3 $ Kloosterman Dirichlet series in the Poisson-summation argument of Deng--Espinosa. The direct local analysis of the Kloosterman series occupies nearly ninety pages in Deng--Espinosa while our treatment here replaces its case-by-case enumeration with a short uniform proof.

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arXiv2026

A degenerate Whittaker criterion for $\mathrm GL_{2n}$

Let $F$ be a non-Archimedean local field. Let $N$ be the unipotent radical of the standard parabolic subgroup of $\mathrm GL_{2n}(F)$ of type $(n,n)$ with fixed nondegenerate additive character $\psi$. For an irreducible admissible representation $\pi$ of $\mathrm GL_{2n}(F)$, a theorem due to Gomez--Gourevitch--Sahi on generalized Whittaker models gives a criterion for the vanishing of the twisted Jacquet module $\pi_{N,\psi}$ in terms of the wave-front set. We translate this orbit-theoretic answer into Langlands--Zelevinsky data: if $\pi=L(\mathfrak m)$, then $\pi_{N,\psi}=0$ if and only if the Zelevinsky dual $\mathfrak m^{\mathrm t}$ contains a segment of length at least $n+1$. We do this in response to a conjecture proposed by D$.$Prasad about the vanishing of $\pi_{N,\psi}$ in terms of the adjoint $L$-function $L(s,\pi\times\pi^\vee)$. We prove that, for every irreducible representation $\pi$, vanishing of $\pi_{N,\psi}$ implies the pole inequalities predicted by D$.$Prasad. However, we show that the converse implication is false by an explicit counterexample for $\mathrm GL_4(F)$. For the generalized Steinberg constituents $v_{P_\beta}^G$ of the principal series containing the trivial representation, we make an explicit calculation of when $\pi_{N,\psi}$ is zero. In particular, for $\mathrm GL_6(F)$, exactly three of the $32$ constituents of such a principal series violate the converse direction of the conjecture proposed by D$.$Prasad.

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arXiv2026

Stability of the exterior cube $\gamma$-factors for $\mathrm{GL}(6)$

We prove the stability of the Langlands-Shahidi local $\gamma$-factor for the exterior cube representation of $\mathrm{GL}_6$. More precisely, if $\pi_1$ and $\pi_2$ are irreducible admissible generic representations of $\mathrm{GL}_6(F)$ with the same central character, then \[ \gamma(s,\pi_1\otimes\chi,\wedge^3,\psi)= \gamma(s,\pi_2\otimes\chi,\wedge^3,\psi) \] for every sufficiently ramified character $\chi$ of $F^\times$, where $\chi$ is regarded as a character of $\mathrm{GL}_6(F)$ through the determinant. The proof uses the realization of the exterior cube representation by the maximal parabolic subgroup of the simply connected group of type $E_6$. We give an explicit description of the relevant geometric quotient $U_M\backslash N'$, compute its invariant measure, and relate Shahidi's partial Bessel functions to partial Bessel integrals on the Levi subgroup. The desired stability then follows from an asymptotic expansion of these partial Bessel integrals and the vanishing of highly ramified Mellin transforms.

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arXiv2026

A trace formula derivation of the functional equation for symmetric square L-functions

This paper revisits the functional equation of the symmetric-square $L$-function from the perspective of trace formulas. We show that, for level-one holomorphic cusp forms and even Hecke--Maass cusp forms, this symmetry can be recovered directly from the Petersson and Kuznetsov trace formulas. Although the functional equation itself is classical, our aim is to provide a concrete and transparent example of how the analytic structure of an $L$-function can emerge from a comparison of the spectral and geometric sides of a trace formula. The argument leads to an averaged reciprocity identity and treats the holomorphic and Maass settings within the same general framework. In this way, the paper offers a simple model for extending trace-formula methods beyond standard $L$-functions.

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arXiv2026

Beyond Endoscopy for $\mathrm{GL}(3, \mathbb{Q})$: Isolation of the residual spectrum via Poisson Summation

We extend to $\mathrm{GL}(3,\mathbb Q)$ the Poisson-summation method developed by Altu\u{g} for $\mathrm{GL}(2,\mathbb Q)$ in Beyond Endoscopy. For a fixed prime and a family of factorizable test functions, we isolate the contribution of the trivial representation from the regular elliptic part of the trace formula and obtain an explicit expansion of \[ \mathrm{I}_{\mathrm{ell}}(f^{p,k})-\mathrm{Tr}(\mathbf{1}(f^{p,k})). \] We rewrite the finite orbital integrals in terms of an overorder zeta function attached to the monogenic cubic orders defined by the characteristic polynomials. Its identification with Yun's zeta function supplies the functional equation needed for an approximate functional equation. A periodicity theorem then makes Poisson summation in the two polynomial coefficients possible. The zero frequency is governed by Kloosterman-type sums and a Dirichlet series that admits a uniform evaluation. The residues at the origin recover the trace of the trivial representation and one third of the trace of the normalized representation induced from the trivial character of the Borel subgroup.

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arXiv2026

Relating Arthur packets of real unitary groups and $p$-adic symplectic and orthogonal groups

We establish an explicit correspondence of certain Arthur packets between real unitary groups and $p$-adic symplectic or orthogonal groups. This allows one to compute Arthur packets of real unitary groups by translating results from the $p$-adic side. A main ingredient in our proof is an explicit relation between Zuckerman's translation functor on the real side and the Jacquet functor on the $p$-adic side. To achieve this, we construct a correspondence of stacks of Langlands parameters with fixed infinitesimal characters between the relevant real and $p$-adic groups. Our approach also allows one to relate the Kazhdan-Lusztig polynomials and the microlocal geometry between real and $p$-adic sides.

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arXiv2025

Distributions of Integral Points and Dedekind Zeta Values

Let $\mathcal{O}$ be the ring of integers for some number field $F$. Let $\chi(x)\in \mathcal{O}[x]$ be a regular monic polynomial of degree $n$. We study the asymptotic count of integral $n\times n$ matrices over $\mathcal{O}$ with the characteristic polynomial $\chi$ and bounded archimedean norm. Previous works establish such an asymptotic with a positive leading constant. Our main result determines this constant in terms of the leading Laurent coefficients at $s=1$ of Dedekind zeta functions attached to orders in $F[x]/(\chi(x))$. The proof combines a refinement of the equi-distribution property of orbits with a reformulation of the counting problem in terms of generalized $\kappa$-orbital integrals. These orbital integrals are then transferred by the endoscopic fundamental lemma and related to zeta functions of orders.

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arXiv2025

Purity of generalized affine Springer fibers from generic planar curve singularities

We prove the cohomological purity of punctual Hilbert schemes of points on generic irreducible planar curve singularities, by constructing an explicit affine paving. Via their identification with generalized $GL_N$-affine Springer fibers attached to the direct sum of the adjoint and standard representations, this establishes a new case of the purity conjecture for generalized affine Springer fibers. The combinatorics of the paving - cell indices and dimensions - are controlled by $(dn,dm)$-Dyck paths extending results of Gorsky-Mazin-Oblomkov on compactified Jacobians. As a byproduct, we also give a simpler proof of their bijection between admissible $(dn,dm)$-invariant subsets and $(dn,dm)$-Dyck paths.

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arXiv2025

On a geometric comparison of representations of complex and $p$-adic $\mathbf{GL}_n$

In this paper, we use geometric methods to study the relations between admissible representations of $\mathbf{GL}_n(\mathbb{C})$ and unramified representations of $\mathbf{GL}_m(\mathbb{Q}_p)$. We show that the geometric relationship between Langlands parameter spaces of $\mathbf{GL}_n(\mathbb{C})$ and $\mathbf{GL}_m(\mathbb{Q}_p)$ constructed by the first named author is compatible with the functor recently defined algebraically by Chan-Wong. We then show that the said relationship intertwines translation functors on representations of $\mathbf{GL}_n(\mathbb{C})$ and partial Bernstein-Zelevinskii derivatives on representations of $\mathbf{GL}_m(\mathbb{Q}_p)$, providing purely geometric counterparts to some results of Chan-Wong. In the sequels, the techniques of this work will be extended to real and $p$-adic classical groups and used to study their Arthur packets.

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arXiv2024

A geometric study of BZ operator on representations of $\mathrm{GL}_n$ over non-archimedean field

In this article, we geometrically study the partial Bernstein-Zelevinsky operator introduced in the author's thesis, which generalizes the original Bernstein-Zelevinsky operator. We relate the partial Bernstein-Zelevinsky operator to the geometric induction of Lusztig and then perform explicit computations in special cases. Finally, we develop a symmetric reduction to the previously mentioned special cases

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arXiv2024

Some geometric consequences of the symmetric reduction

In this article, we derived some consequences to the symmetrization process developed in \cite{Deng23}. This consists a geometric derivation of part of the properties which uniquely determines the Kazhdan-Lusztig polynomials of type $A_n$ as well as an interpretations of the last property by the decomposition theorem of \cite{BBD}. Finally, the relation of geometry of graded nilpotent classes to Parabolic Kazhdan-Lusztig polynomials is discussed for the purpose of later applications to Zelevinsky's derivatives.

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arXiv2023

Stability of Rankin-Selberg local $\gamma$-factors for split classical groups: the symplectic case

Given a split classical group of symplectic type and a split general linear group over a local field $F$, we use Langlands-Shahidi method to construct their Rankin-Selberg local $\gamma$-factors and prove the corresponding analytic stability for generic representations. The idea generalizes the work of J. Cogdell, F. Shahidi, T.-L. Tsai in 2017 and D. She in 2023 in the study of asymptotic behaviors of partial Bessel functions. Different from the known cases, suppose $P=MN$ is the maximal parabolic subgroup with Levi component $M\simeq \mathrm{GL}_r\times\mathrm{Sp}_{2m}$ that defines the local factors, the action of the maximal unipotent subgroup of $M$ on $N$ have non-trivial stabilizers, and the space of integration for the corresponding local coefficient is no longer isomorphic to a torus. We will separate its toric part out in our cases and show that it plays the same role as the torus over which the integral representing the local coefficient is taken in the known cases. This is a new phenomenon with sufficient generality and we believe that it may provide us with a possible direction towards a uniform proof of stability of Langlands-Shahidi $\gamma$-factors in our future work.

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arXiv2021

Poisson summation for Hankel transforms

In this article we study the Poisson summation for Hankel transform in the sense of Braverman-Kazhdan-Ngo in the special case of $L$-embedding $\rho: GL_1\rightarrow GL_2$. We view such a summation formula as the generalization of the classical Voronoi summation formula.

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arXiv2020

The characteristic cycles and semi-canonical bases on type $A$ quiver variety

In this article we study a conjecture of Geiss-Leclerc-Schr{\"o}er, which is an analogue of a classical conjecture of Lusztig in the Weyl group case. It concerns the relation between canonical basis and semi-canonical basis through the characteristic cycles. We formulate an approach to this conjecture and prove it for type $A_2$ quiver. In general type A case, we reduce the conjecture to show that certain nearby cycles have vanishing Euler characteristic.

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arXiv2019

Torsions in Cohomology of $\text{SL}_2(\mathbb{Z})$ and Congruence of Modular Forms

We describe torsion classes in the first cohomology group of $\text{SL}_2(\mathbb{Z})$. In particular, we obtain generalized Dickson's invariants for p-power polynomial rings. Secondly, we describe torsion classes in the zero-th homology group of $\text{SL}_2(\mathbb{Z})$ as a module over the torsion invariants. As application, we obtain various congruences between cuspidal forms of level one and Eisenstein series.

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arXiv2019

Symmetrization of representations of $GL_N$

In this article, we develop a process to symmetrize the irreducible admissible representation of $GL_N(\mathbb{Q}_p)$, as a consequence we obtain a more geometric understanding of the coefficient $m(\mathbf{b}, \mathbf{a})$ appearing in the decomposition of parabolic inductions, which allows us to prove a conjecture posed by Zelevinsky.

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