Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2105.08754

A weighted invariant trace formula

Tian An Wong
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper begins a new approach to the $r$-trace formula, without removing the nontempered contribution to the spectral side. We first establish an invariant trace formula whose discrete spectral terms are weighted by automorphic $L$-functions. This involves extending the results of Finis, Lapid, and Müller on the continuity of the coarse expansion of Arthur's noninvariant trace formula to the refined expansion, and then to the invariant trace formula, while incorporating the use of basic functions at unramified places.

Explore related subjects

math.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tian An Wong. 2022-02-08. A weighted invariant trace formula. https://arxiv.org/abs/2105.08754

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the first Hochschild cohomology of finite dimensional quiver algebras under gluing idempotents

We compare the Lie algebra structures of the first Hochschild cohomology groups of a quiver algebra $A$ and a radical embedding $B$ obtained by gluing two idempotents of $A$. Under a mild assumption, we show that the first Hochschild cohomology groups of $A$ and $B$ are either isomorphic as Lie algebras or they differ by a one-dimensional Lie ideal. In particular, in the case of stable equivalences obtained by gluing a source and a sink vertex, we prove that either the first Hochschild cohomology groups of $A$ and $B$ are isomorphic or $HH^1(B)$ is a central extension of $HH^1(A)$ by a one-dimensional ideal. As a consequence, we obtain a new invariant under stable equivalences induced by gluing a source and a sink. We also compare the dimensions of $HH^1(A)$ and $HH^1(B)$, as well as the centers of $A$ and $B$, when gluing two arbitrary idempotents.

math.RT↗
arXiv2026

From quantum difference equation to Dubrovin connection of affine type A quiver varieties

This is the continuation of the article \cite{Z23}. In this article we will give a detailed analysis of the quantum difference equation of the equivariant $K$-theory of the affine type $A$ quiver varieties. We will give a good representation of the quantum difference operator $\mathbf{M}_{\mathcal{L}}(z)$ such that the monodromy operator $\mathbf{B}_{\mathbf{m}}(z)$in the formula can be written in the $U_{q}(\mathfrak{sl}_2)$-form or in the $U_{q}(\hat{\mathfrak{gl}}_1)$-form. We also give the detailed analysis of the connection matrix for the quantum difference equation in the nodal limit $p\rightarrow0$. Using these two results, we prove that the degeneration limit of the quantum difference equation is the Dubrovin connection for the quantum cohomology of the affine type $A$ quiver varieties, and the monodromy representation for the Dubrovin connection is generated by the monodromy operators $\mathbf{B}_{\mathbf{m}}$.

math.RT↗
arXiv2026

Hessian Defect, Compatibility Degree, and Canonical Decomposition

We express the difference between dimension and denominator vectors as the rank defect of a Hessian differential. For a very general Jacobi-finite potential, this defect vanishes on a general representation in a principal component precisely when the positive-length cycles at the vertex act trivially. The associated Hessian vectors satisfy tropical $X$-mutation and Auslander--Reiten translation. We also give two distinct cluster variables with the same denominator vector but different Hessian vectors. Their Hom pairing extends ordered denominator compatibility, and its negative part computes the canonical multiplicity of any extended-reachable indecomposable class in an arbitrary presentation weight.

math.RT↗