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Roberto J. Miatello

Publications and source records attributed to Roberto J. Miatello.

11 recordsLinked to original sources
arXiv2021

Representations of the unitary group SU(2,1) in Fourier term modules

We study Fourier term modules on $\mathrm{SU}(2,1)$, which are the modules arising in Fourier expansions of automorphic forms. Maximal unipotent subgroups $N$ of $\mathrm{SU}(2,1)$ are non-abelian, and we consider the ``abelian'' Fourier term modules connected to characters of $N$, and also the ``non-abelian'' modules described with theta functions. Poincaré series for $\mathrm{SU}(2,1)$ have in general exponential growth. To deal with such generalized automorphic forms we allow exponential growth for the functions in Fourier term modules. We give a complete description of the submodule structure of all Fourier term modules, and discuss the consequences for Fourier expansions of automorphic forms.

math.RT↗
arXiv2021

Generalized Poincaré series for $\mathrm{SU}(2,1)$

We define and study 'non-abelian' Poincaré series for the group $G=\mathrm{SU} (2,1)$, i.e. Poincaré series attached to a Stone-Von Neumann representation of the unipotent subgroup $N$ of $G$. Such Poincaré series have in general exponential growth. In this study we use results on abelian and non-abelian Fourier term modules obtained in arXiv:1912.01334. We compute the inner product of truncations of these series and those associated to unitary characters of $N$ with square integrable automorphic forms, in connection with their Fourier expansions. As a consequence, we obtain general completeness results that, in particular, generalize those valid for the classical holomorphic (and antiholomorphic) Poincaré series for $\mathrm{SL}(2,\mathbb{R})$.

math.NT↗
arXiv2021

Strong representation equivalence for compact symmetric spaces of real rank one

Let $G/K$ be a simply connected compact irreducible symmetric space of real rank one. For each $K$-type $τ$ we compare the notions of $τ$-representation equivalence with $τ$-isospectrality. We exhibit infinitely many $K$-types $τ$ so that, for arbitrary discrete subgroups $Γ$ and $Γ'$ of $G$, if the multiplicities of $λ$ in the spectra of the Laplace operators acting on sections of the induced $τ$-vector bundles over $Γ\backslash G/K$ and $Γ'\backslash G/K$ agree for all but finitely many $λ$, then $Γ$ and $Γ'$ are $τ$-representation equivalent in $G$ (i.e.\ $\dim \operatorname{Hom}_G(V_π, L^2(Γ\backslash G))=\dim \operatorname{Hom}_G(V_π, L^2(Γ'\backslash G))$ for all $π\in \widehat G$ satisfying $\operatorname{Hom}_K(V_τ,V_π)\neq0$). In particular $Γ\backslash G/K$ and $Γ'\backslash G/K$ are $τ$-isospectral (i.e.\ the multiplicities agree for all $λ$). We specially study the case of $p$-form representations, i.e. the irreducible subrepresentations $τ$ of the representation $τ_p$ of $K$ on the $p$-exterior power of the complexified cotangent bundle $\bigwedge^p T_{\mathbb C}^*M$. We show that for such $τ$, in most cases $τ$-isospectrality implies $τ$-representation equivalence. We construct an explicit counter-example for $G/K= \operatorname{SO}(4n)/ \operatorname{SO}(4n-1)\simeq S^{4n-1}$.

math.DG↗
arXiv2020

Joint distribution of eigenvalues of Hecke and Casimir operators for Hilbert Maass forms

Let $F$ be a totally real number field, $\mathcal{O}_{F}$ the ring of integers, $\mathfrak a$ and $\mathfrak I$ integral ideals and let $χ$ a character of $\mathbb{A}_F^\times/F^\times$. For each prime ideal $\mathfrak{p}$ in $\mathcal{O}_{F}$, $\mathfrak{p}\nmid \mathfrak{I}$ let $T_{\mathfrak{p}}$ be the Hecke operator acting on the space of Maass cusp forms on $L^2(\mathrm{GL}_{2}(F) \backslash \mathrm{GL}_{2}(\mathbb{A}_F))$. In this paper we investigate the distribution of joint eigenvalues of the Hecke operators $T_{\mathfrak{p}}$ and of the Casimir operators $C_{j}$ in each archimedean component of $F$, for $1\le j \le d$. Summarily, we prove that given a family of expanding compact subsets $Ω_{t}$ of $\mathbb{R}^{d}$ as $t \rightarrow \infty$, and an interval $I_{\mathfrak{p}} \subseteq [-2,2]$, then, if $\mathfrak{p} \nmid \mathfrak{I}$ is a square in the narrow class group of $F$, there are infinitely many automorphic forms having eigenvalues of $T_{\mathfrak{p}}$ in $I_{\mathfrak{p}}$, distributed on $I_{\mathfrak{p}}$ according to a polynomial multiple of the Sato-Tate measure and having their Casimir eigenvalues in the region $Ω_{t}$, distributed according to the Plancherel measure.

math.NT↗
arXiv2020

Strong multiplicity one theorems for locally homogeneous spaces of compact type

Let $G$ be a compact connected semisimple Lie group, let $K$ be a closed subgroup of $G$, let $Γ$ be a finite subgroup of $G$, and let $τ$ be a finite-dimensional representation of $K$. For $π$ in the unitary dual $\widehat G$ of $G$, denote by $n_Γ(π)$ its multiplicity in $L^2(Γ\backslash G)$. We prove a strong multiplicity one theorem in the spirit of Bhagwat and Rajan, for the $n_Γ(π)$ for $π$ in the set $\widehat G_τ$ of irreducible $τ$-spherical representations of $G$. More precisely, for $Γ$ and $Γ'$ finite subgroups of $G$, we prove that if $n_Γ(π)= n_{Γ'}(π)$ for all but finitely many $π\in \widehat G_τ$, then $Γ$ and $Γ'$ are $τ$-representation equivalent, that is, $n_Γ(π)=n_{Γ'}(π)$ for all $π\in \widehat G_τ$. Moreover, when $\widehat G_τ$ can be written as a finite union of strings of representations, we prove a finite version of the above result. For any finite subset $\widehat {F}_τ$ of $\widehat G_τ$ verifying some mild conditions, the values of the $n_Γ(π)$ for $π\in\widehat F_τ$ determine the $n_Γ(π)$'s for all $π\in \widehat G_τ$. In particular, for two finite subgroups $Γ$ and $Γ'$ of $G$, if $n_Γ(π) = n_{Γ'}(π)$ for all $π\in \widehat F_τ$ then the equality holds for every $π\in \widehat G_τ$. We use algebraic methods involving generating functions and some facts from the representation theory of $G$.

math.RT↗
arXiv2019

Recent results on the spectra of lens spaces

In this paper we report on recent results by several authors, on the spectral theory of lens spaces and orbifolds and similar locally symmetric spaces of rank one. Most of these results are related to those obtained by the authors in [IMRN (2016), 1054--1089], where the spectra of lens spaces were described in terms of the one-norm spectrum of a naturally associated congruence lattice. As a consequence, the first examples of Riemannian manifolds isospectral on $p$-forms for all $p$ but not strongly isospectral were constructed. We also give a new elementary proof in the case of the spectrum on functions. In this proof, representation theory of compact Lie groups is avoided and replaced by the use of Molien's formula and a manipulation of the one-norm generating function associated to a congruence lattice. In the last four sections we present several recent results, open problems and conjectures on the subject.

math.DG↗
arXiv2015

Non-strongly isospectral spherical space forms

In this paper we describe recent results on explicit construction of lens spaces that are not strongly isospectral, yet they are isospectral on $p$-forms for every $p$. Such examples cannot be obtained by the Sunada method. We also discuss related results, emphasizing on significant classical work of Ikeda on isospectral lens spaces, via a thorough study of the associated generating functions.

math.DG↗
arXiv2015

Spectra of lens spaces from 1-norm spectra of congruence lattices

To every $n$-dimensional lens space $L$, we associate a congruence lattice $\mathcal L$ in $\mathbb Z^m$, with $n=2m-1$ and we prove a formula relating the multiplicities of Hodge-Laplace eigenvalues on $L$ with the number of lattice elements of a given $\|\cdot\|_1$-length in $\mathcal L$. As a consequence, we show that two lens spaces are isospectral on functions (resp.\ isospectral on $p$-forms for every $p$) if and only if the associated congruence lattices are $\|\cdot\|_1$-isospectral (resp.\ $\|\cdot\|_1$-isospectral plus a geometric condition). Using this fact, we give, for every dimension $n\ge 5$, infinitely many examples of Riemannian manifolds that are isospectral on every level $p$ and are not strongly isospectral.

math.DG↗
arXiv2012

Representation equivalence and p-Spectrum of constant curvature space forms

We study the $p$-spectrum of a locally symmetric space of constant curvature $Γ\backslash X$, in connection with the right regular representation of the full isometry group $G$ of $X$ on $L^2(Γ\backslash G)_{τ_p}$, where $τ_p$ is the complexified $p$-exterior representation of $\mathrm{O}(n)$ on $\bigwedge^p(\mathbb{R}^n)_\mathbb{C}$. We give an expression of the multiplicity $d_λ(p,Γ)$ of the eigenvalues of the $p$-Hodge-Laplace operator in terms of multiplicities $n_Γ(π)$ of specific irreducible unitary representations of $G$. As a consequence, we extend results of Pesce for the spectrum on functions to the $p$-spectrum of the Hodge-Laplace operator on $p$-forms of $Γ\backslash X$, and we compare $p$-isospectrality with $τ_p$-equivalence for $0\leq p\leq n$. For spherical space forms, we show that $τ$-isospectrality implies $τ$-equivalence for a class of $τ$'s that includes the case $τ=τ_p$. Furthermore we prove that $p-1$ and $p+1$-isospectral implies $p$-isospectral. For nonpositive curvature space forms, we give examples showing that $p$-isospectrality is far from implying $τ_p$-equivalence, but a variant of Pesce's result remains true. Namely, for each fixed $p$, $q$-isospectrality for every $0\leq q\leq p$ implies $τ_q$-equivalence for every $0\leq q\leq p$. As a byproduct of the methods we obtain several results relating $p$-isospectrality with $τ_p$-equivalence.

math.SP↗
arXiv2011

Strongly isospectral manifolds with nonisomorphic cohomology rings

For any $n\geq 7$, $k\geq 3$, we give pairs of compact flat $n$-manifolds $M, M'$ with holonomy groups $\mathbb Z_2^k$, that are strongly isospectral, hence isospectral on $p$-forms for all values of $p$, having nonisomorphic cohomology rings. Moreover, if $n$ is even, $M$ is Kähler while $M'$ is not. Furthermore, with the help of a computer program we show the existence of large Sunada isospectral families; for instance, for $n=24$ and $k=3$ there is a family of eight compact flat manifolds (four of them Kähler) having very different cohomology rings. In particular, the cardinalities of the sets of primitive forms are different for all manifolds.

math.DG↗