arXiv subjects

Michele Zordan

Publications and source records attributed to Michele Zordan.

6 recordsLinked to original sources

arXiv2024

Rationality of representation zeta functions of compact $p$-adic analytic groups

We prove that for any FAb compact $p$-adic analytic group $G$, its representation zeta function is a finite sum of terms $n_{i}^{-s}f_{i}(p^{-s})$, where $n_{i}$ are natural numbers and $f_{i}(t)\in\mathbb{Q}(t)$ are rational functions. Meromorphic continuation and rationality of the abscissa of the zeta function follow as corollaries. If $G$ is moreover a pro-$p$ group, we prove that its representation zeta function is rational in $p^{-s}$. These results were proved by Jaikin-Zapirain for $p>2$ or for $G$ uniform and pro-$2$, respectively. We give a new proof which avoids the Kirillov orbit method and works for all $p$. First part of arXiv:2007.10694, second part uploaded as a separate paper.

math.GR↗

arXiv2024

Rationality of twist representation zeta functions of compact $p$-adic analytic groups

We prove that for any twist rigid compact $p$-adic analytic group $G$, its twist representation zeta function is a finite sum of terms $n_{i}^{-s}f_{i}(p^{-s})$, where $n_{i}$ are natural numbers and $f_{i}(t)\in\mathbb{Q}(t)$ are rational functions. Meromorphic continuation and rationality of the abscissa of the zeta function follow as corollaries. If $G$ is moreover a pro-$p$ group, we prove that its twist representation zeta function is rational in $p^{-s}$. To establish these results we develop a Clifford theory for twist isoclasses of representations, including a new cohomological invariant of a twist isoclass. Second part of arXiv:2007.10694.

math.GR↗

arXiv2022

Univariate and bivariate zeta functions of unipotent group schemes of type $G$

We compute the representation and class counting zeta functions for a family of torsion-free finitely generated nilpotent groups of nilpotency class $2$. These groups arise from a generalisation of one the families of unipotent groups schemes treated by Stasinski and Voll, and Lins. The univariate zeta functions are obtained by specialising the respective bivariate zeta functions defined by Lins. These are also used to deduce a formula for a joint distribution on Weyl groups of type $B$.

math.GR↗

arXiv2018

On representation zeta functions for special linear groups

We prove that the numbers of irreducible n-dimensional complex continuous representations of the special linear groups over p-adic integers grow slower than the square of n. We deduce that the abscissas of convergence of the representation zeta functions of the special linear groups over the ring of integers are bounded above by 2. In order to show these results we prove also that if G is a connected, simply connected, semi-simple algebraic group defined over the field of rational numbers, then the G-representation variety of the fundamental group of a compact Riemann surface of genus n has rational singularities if and only if the G-character variety has rational singularities.

math.AG↗

arXiv2018

Adjoint Orbits of Matrix Groups over Finite Quotients of Compact Discrete Valuation Rings and Representation Zeta Functions

This paper gives methods to describe the adjoint orbits of $\mathbf{G}(\mathfrak{o}_r)$ on $\mathrm{Lie}(\mathbf{G})(\mathfrak{o}_r)$ where $\mathfrak{o}_r=\mathfrak{o}/\mathfrak{p}^r$ ($r\in\mathbb{N}$) is a finite quotient of the localization $\mathfrak{o}$ of the ring of integers of a number field at a prime ideal $\mathfrak{p}$ and $\mathbf{G}$ is a closed $\mathbb{Z}$-subgroup scheme of $\mathrm{GL}_{n}$ for an $n\in\mathbb{N}$ and such that the Lie ring $\mathrm{Lie}(\mathbf{G})(\mathfrak{o})$ is quadratic.. The main result is a classification of the adjoint orbits in $\mathrm{Lie}(\mathbf{G})(\mathfrak{o}_{r+1})$ whose reduction $\bmod\,\mathfrak{p}^{r}$ contains $a\in\mathrm{Lie}(\mathbf{G})(\mathfrak{o}_r)$ in terms of the reduction $\bmod\mathfrak{p}$ of the stabilizer of $a$ for the $\mathbf{G}(\mathfrak{o}_r)$-adjoint action. As an application, this result is then used to compute the representation zeta function of the principal congruence subgroups of $\mathrm{SL}_{3}(\mathfrak{o})$.

math.GR↗

arXiv2017

Poincaré series of Lie lattices and representation zeta functions of arithmetic groups

We compute explicit formulae for Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of potent and saturable principal congruence subgroups of $\mathrm{SL}_4^m(\mathfrak{o})$ ($m\in\mathbb{N}$) for $\mathfrak{o}$ a compact DVR of characteristic $0$ and odd residue field characteristic. In doing so we develop a novel method for computing Poincaré series associated with commutator matrices of $\mathfrak{o}$-Lie lattices with finite abelianization and whose rank-loci enjoy an additional smoothness property. We give explicit formulae for the abscissa of convergence of the representation zeta functions of potent and saturable FAb $p$-adic analytic groups whose associated Lie lattices satisfy the hypotheses of the aforementioned method. As a by-product of our computations we find that not all $4\times 4$ traceless matrices over a finite quotient of $\mathfrak{o}$ admit shadow-preserving lifts, thus disproving that smooth loci of constant centralizer dimension in $\mathfrak{sl}_4(\mathbb{C})$ ensure presence of shadow-preserving lifts for almost all primes as suggested in a previous paper by Avni, Klopsch, Onn and Voll.

math.GR↗