arXiv subjects

Kaustabh Mondal

Publications and source records attributed to Kaustabh Mondal.

4 recordsLinked to original sources

arXiv2026

Potential semistability of Finite height Galois representations: Relative case

Let $K$ be a $p$-adic field. We define the notion of finite height for an étale $\mathbb{Z}_p$-local system on a smooth adic space $\mathcal{X}$ over $K$ with semistable reduction. Using analytic prismatic $F$-crystals and purity results of Du-Liu-Moon-Shimizu (arXiv:2404.19603), we prove that if an étale $\mathbb{Z}_p$-local system over $\mathcal{X}$ is of finite height then its pullback along a finite étale cover of $\mathcal{X}$ is semistable. This answers a question of Tong Liu in the relative setting.

math.NT↗

arXiv2025

Representation Equivalence of Lattices in Lie Groups

Let $Γ_1$ and $Γ_2$ be two lattices of finite covolume in a semisimple Lie group $G$. We prove a spectral rigidity result for the representation spectra of the right regular representations $L^2(Γ_1 \backslash G)$ and $L^2(Γ_2 \backslash G)$ of $G$. This can be thought of as an analogue of the strong multiplicity one theorem and it generalises a result by the first author and Rajan to the case of non-uniform lattices.

math.RT↗

arXiv2025

A finite Linear Dependence of Discrete Series Multiplicities

Let $G$ be a connected semisimple simply connected Lie group with a compact Cartan subgroup and let $Γ$ be a uniform lattice in $G$. Let $\widehat{G}_d$ denote the set of equivalence classes of unitary discrete series representations of $G$. We prove that for any finite subset of $\widehat{G}_d$ satisfying a certain condition, the associated finite set of discrete series multiplicities in $L^2(Γ\backslash G)$ determines all discrete series multiplicities in $L^2(Γ\backslash G)$. This allows us to obtain a refinement of the strong multiplicity one result for discrete series representations. As an application, we deduce that for two given levels, the equality of the dimensions of the spaces of cusp forms over a suitable finite set of weights implies the equality of the dimensions of the spaces of cusp forms for all weights.

math.RT↗

arXiv2024

On Infinitesimal $τ$-Isospectrality of Locally Symmetric Spaces

Let $(τ, V_τ)$ be a finite dimensional representation of a maximal compact subgroup $K$ of a connected non-compact semisimple Lie group $G$, and let $Γ$ be a uniform torsion-free lattice in $G$. We obtain an infinitesimal version of the celebrated Matsushima-Murakami formula, which relates the dimension of the space of automorphic forms associated to $τ$ and multiplicities of irreducible $τ^\vee$-spherical spectra in $L^2(Γ\backslash G)$. This result gives a promising tool to study the joint spectra of all central operators on the homogenous bundle associated to the locally symmetric space and hence its infinitesimal $τ$-isospectrality. Along with this we prove that the almost equality of $τ$-spherical spectra of two lattices assures the equality of their $τ$-spherical spectra.

math.RT↗