arXiv subjects

Hiroyoshi Tamori

Publications and source records attributed to Hiroyoshi Tamori.

6 recordsLinked to original sources

arXiv2026

Root lattices over totally real fields

A root lattice is a finite rank $\mathbb{Z}$-lattice generated by elements $x$ satisfying $x\cdot x=2$. It is well-known that the root lattices have an $ADE$ classification and they play a prominent role in the study of even unimodular lattices. The notion of root lattices can be naturally generalized to lattices over the ring of integers $\mathcal{O}$ of a totally real field $K$. In the case where $K$ is a real quadratic field, such lattices were classified by Mimura in 1979, and this classification has been used by several researchers in the study of even unimodular $\mathcal{O}$-lattices. In this paper, we extend this classification to arbitrary totally real fields. The irreducible root lattices of rank greater than $2$ are indexed by finite Coxeter systems. All the rank $2$ root lattices are realized as orders in quadratic extensions of $K$ and their classification requires some technique from algebraic number theory.

math.CO↗

arXiv2025

Strichartz Estimates for the $(k,a)$-Generalized Laguerre Operators

In this paper, we prove Strichartz estimates for the $(k,a)$-generalized Laguerre operators $a^{-1}\bigl(-|x|^{2-a}Δ_k+|x|^a\bigr)$ which were introduced by Ben Sa\"ıd-Kobayashi-Orsted, and for the operators $|x|^{2-a}Δ_k$. Here $k$ denotes a non-negative multiplicity function for the Dunkl Laplacian $Δ_k$ and $a$ denotes a positive real number satisfying certain conditions. The cases $a=1,2$ were studied previously. We consider more general cases here. The proof depends on symbol-type estimates of special functions and a discrete analog of the stationary phase theorem inspired by the work of Ionescu-Jerison.

math.AP↗

arXiv2023

Epsilon dichotomy for linear models: the Archimedean case

Let $G=\mathrm{GL}_{2n}(\mathbb{R})$ or $G=\mathrm{GL}_n(\mathbb{H})$ and $H=\mathrm{GL}_n(\mathbb{C})$ regarded as a subgroup of $G$. Here, $\mathbb{H}$ is the quaternion division algebra over $\mathbb{R}$. For a character $χ$ on $\mathbb{C}^\times$, we say that an irreducible smooth admissible moderate growth representation $π$ of $G$ is $χ_H$-distinguished if $\mathrm{Hom}_H(π, χ\circ\det_H)\neq0$. We compute the root number of a $χ_H$-distinguished representation $π$ twisted by the representation induced from $χ$. This proves an Archimedean analogue of the conjecture by Prasad and Takloo-Bighash (J. Reine Angew. Math., 2011). The proof is based on the analysis of the contribution of $H$-orbits in a flag manifold of $G$ to the Schwartz homology of principal series representations. A large part of the argument is developed for general real reductive groups of inner type. In particular, we prove that the Schwartz homology $H_\ast(H, π\otimesχ)$ is finite-dimensional and hence it is Hausdorff for a reductive symmetric pair $(G, H)$ and a finite-dimensional representation $χ$ of $H$.

math.NT↗

arXiv2022

Iwahori-Hecke algebra and unramified local L-functions

In this paper, we compute the Hecke action of a certain test function on the space of an unramified principal series of a connected reductive group over a non-archimedean local field by using the theory of Iwahori--Hecke algebra. As an application, we obtain a new expression of the local L-functions of unramified representations.

math.NT↗

arXiv2021

Classification of irreducible $(\mathfrak{g},\mathfrak{k})$-modules associated to the ideals of minimal nilpotent orbits for simple Lie groups of type $A$

We classify completely prime primitive ideals whose associated varieties are the closure of the minimal nilpotent orbit of $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, and classify irreducible $(\mathfrak{g},\mathfrak{k})$-modules which have those ideals as annihilators. Moreover, we irreducibly decompose them as $\mathfrak{k}$-modules.

math.RT↗

arXiv2018

Classification of minimal representations of real simple Lie groups

Based on an idea in [Gan--Savin, Represent. Theory (2005)], we give a classification of minimal representations of connected simple real Lie groups not of type $A$. Actually, we prove that there exist no new minimal representations up to infinitesimal equivalence.

math.RT↗