Searcharxiv⌕ Search

arXiv subjects

Fabian Januszewski

Publications and source records attributed to Fabian Januszewski.

16 recordsLinked to original sources
arXiv2025

Rational quiver representations: tame and wild

We study representation finite $K$-rational quivers over fields of characteristic $0$ and their indecomposable representations, exploiting that all Brauer obstructions for descent of representations are trivial in this case. Contrasting the tame case, we give an example of a simple quiver of wild representation type, where we realize every possible Brauer obstruction of a given Galois extension $L/K$ in the category of quiver representations over $L$.

math.RT↗
arXiv2025

Rational structures on quivers and a generalization of Gelfand's equivalence

We introduce the notion of rational structure on a quiver and associated representations to establish a coherent framework for studying quiver representations in separable field extensions. This notion is linked to a refinement of the notion of $K$-species, which we term \'etale $K$-species: We establish a categorical anti-equivalence between the category of $K$-rational quivers and that of \'etale $K$-species, which extends to an equivalence of their respective representation categories. For $K$-rational quivers there is a canonical notion of base change, which suggests a corresponding notion of base change for (\'etale) $K$-species which we elaborate. As a primary application, we generalize Gelfand's celebrated equivalence between certain blocks of Harish-Chandra modules for $\mathrm{SL}_2(\mathbb{R})$ and representations of the Gelfand quiver to a rational setting. To this end, we define a $\mathbb{Q}$-rational structure on the Gelfand quiver and its representations. A key technical tool, which we call unipotent stabilization, is developed to construct the functor from certain rational Harish-Chandra modules to nilpotent rational quiver representations. We prove that this functor is an equivalence. A similar result is established for the cyclic quiver. A notable consequence of this rational framework is that the defining relation of the Gelfand quiver becomes superfluous when working over fields not containing $\sqrt{-1}$. This allows us to recast our results in the language of $\mathbb{Q}$-species without relations.

math.RT↗
arXiv2025

Hausdorffness of certain nilpotent cohomology spaces

Let $(\pi,V)$ be a smooth representation of a compact Lie group $G$ on a quasi-complete locally convex complex topological vector space. We show that the Lie algebra cohomology space $\mathrm{H} ^\bullet(\mathfrak{u}, V)$ and the Lie algebra homology space $\mathrm{H}_\bullet(\mathfrak{u}, V)$ are both Hausdorff, where $\mathfrak{u}$ is the nilpotent radical of a parabolic subalgebra of the complexified Lie algebra $\mathfrak{g}$ of $G$.

math.RT↗
arXiv2024

Locally algebraic representations and integral structures on the cohomology of arithmetic groups

This paper introduces the notion of locally algebraic representations and corresponding sheaves in the context of the cohomology of arithmetic groups. These representations are of relevance for the study of integral structures and special values of cohomological automorphic representations, as well as corresponding period relations. We introduce and investigate related concepts such as locally algebraic $({\mathfrak g},K)$-modules and cohomological types of automorphic representations. Applying the recently developed theory of tdos and twisted $\mathcal D$-modules over schemes by Hayashi and the author, we establish the existence of canonical global $1/N$-integral structures on spaces of automorphic cusp forms. As an application, we define canonical periods attached to regular algebraic automorphic representations, potentially related to the action of Venkatesh's derived Hecke algebra on cuspidal cohomology.

math.NT↗
arXiv2021

Towards a tensor-classification of Harish-Chandra modules: The case ${\rm SL}_2(\mathbb R)$

We consider the category of Harish-Chandra modules for ${\rm SL}_2(\mathbb R)$ as a module over the category of finite-dimensional representations of ${\rm SL}(2)$ with respect to the tensor product. In this note we use classical results about principal series to obtain a classification of $\otimes$-submodules of the category of Harish-Chandra modules of ${\rm SL}_2(\mathbb R)$. The resulting classification recovers the classical classification of irreducible Harish-Chandra modules. Our methods are expected to generalize to arbitrary reductive pairs and more general base fields.

math.RT↗
arXiv2020

$L$-functions of ${\mathrm{GL}}(2n):$ $p$-adic properties and non-vanishing of twists

The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $Π$ of $\mathrm{GL}(2n)$ over a totally real field $F$ admitting a Shalika model. We use a modular symbol approach, along the global lines of the work of Ash and Ginzburg, but our results are more definitive since we draw heavily upon the methods used in the recent and separate works of all the three authors. By construction our $p$-adic $L$-functions are distributions on the Galois group of the maximal abelian extension of $F$ unramified outside $p\infty$. Moreover we work under a weaker Panchishkine type condition on $Π_p$ rather than the full ordinariness condition. Finally, we prove the so-called Manin relations between the $p$-adic $L$-functions at all critical points. This has the striking consequence that, given a unitary $Π$ whose standard $L$-function admits at least two critical points, and given a prime $p$ such that $Π_p$ is ordinary, the central critical value $L(\tfrac12, Π\otimesχ)$ is non-zero for all except finitely many Dirichlet characters $χ$ of $p$-power conductor.

math.NT↗
arXiv2018

Families of twisted $\mathcal D$-modules and arithmetic models of Harish-Chandra modules

We develop a theory of tdos and twisted $\mathcal D$-modules over general base schemes with a focus on functorial aspects. In particular, we introduce a flat base change functor and establish its compatibility with globalization and direct image functors. We also study forms of closed $K$-orbits of $\theta$-stable parabolic subgroups in the total flag variety. We apply these two developed theories to give a geometric construction of half-integral models of cohomologically induced modules. With a view towards arithmetic applications, we further demonstrate desirable properties of the constructed half-integral models, such as projectivity over the base and torsion-free relative Lie algebra cohomology.

math.RT↗
arXiv2017

On Period Relations for Automorphic L-functions I

This paper is the first in a series of two dedicated to the study of period relations of the type $$ L(\frac{1}{2}+k,Π)\;\in\;(2πi)^{d\cdot k}Ω_{(-1)^k}{\mathbb Q}(Π),\quad \frac{1}{2}+k\;\text{critical}, $$ for certain automorphic representations $Π$ of a reductive group $G.$ In this paper we discuss the case $G={\mathrm{GL}}(n+1)\times{\mathrm{GL}}(n).$ The case $G={\mathrm{GL}}(2n)$ is discussed in part two. Our method is representation-theoretic and relies on the author's recent results on global rational structures on automorphic representations. We show that the above period relations are intimately related to the field of definition of the global representation $Π$ under consideration. The new period relations we prove are in accordance with Deligne's Conjecture on special values of $L$-functions and we expect our method to apply to other cases as well.

math.NT↗
arXiv2017

Non-abelian $p$-adic Rankin-Selberg $L$-functions and non-vanishing of central $L$-values

We prove new congruences between special values of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over arbitrary number fields. This allows us to control the behavior of $p$-adic $L$-functions under Tate twists and to prove the existence of non-abelian $p$-adic $L$-functions for Hida families on $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$. As an application, we prove strong non-vanishing results for central $L$-values: We give sufficient local conditions for twisted central Rankin-Selberg $L$-values to be generically non-zero.

math.NT↗
arXiv2017

Rational Structures on Automorphic Representations

This paper proves the existence of global rational structures on spaces of cusp forms of general reductive groups. We identify cases where the constructed rational structures are optimal, which includes the case of GL($n$). As an application, we deduce the existence of a natural set of periods attached to cuspidal automorphic representations of GL($n$). This has consequences for the arithmetic of special values of $L$-functions that we discuss in subsequent articles. In the course of proving our results, we lay the foundations for a general theory of Harish-Chandra modules over arbitrary fields of characteristic $0$. In this context, a rational character theory, translation functors and an equivariant theory of cohomological induction are developed. We also study descent problems for Harish-Chandra modules in quadratic extensions, where we obtain a complete theory over number fields.

math.NT↗
arXiv2016

On Period Relations for Automorphic L-functions II

We study Hecke algebras for pairs $({\mathfrak g},K)$ over arbitrary fields $E$ of characteristic $0$, define the Bernstein functor and give another definition of the Zuckerman functor over $E$. Building on this and the author's previous work on rational structures on automorphic representations, we show that hard duality remains valid over $E$ and apply this result to the study of rationality properties of Sun's cohomologically induced functionals. Our main application are period relations for the special values of standard $L$-functions of automorphic representations of $\mathrm{GL}(2n)$ admitting Shalika models.

math.NT↗
arXiv2014

On p-adic L-functions for ${\rm GL}(n)\times{\rm GL}(n-1)$ over totally real fields

We refine and extend previous constructions of $p$-adic $L$-functions for Rankin-Selberg convolutions on $\GL(n)\times\GL(n-1)$ for regular algebraic representations over totally real fields. We also prove an intrinsic functional equation for this $p$-adic $L$-function, which will be of interest in further study of its arithmetic properties.

math.NT↗
arXiv2013

Algebraic Characters for Harish-Chandra modules

We give a cohomological treatment of a character theory for (g,K)-modules. This leads to a nice formalism extending to large categories of not necessarily admissible (g,K)-modules. Due to results of Hecht, Schmid and Vogan the classical results of Harish-Chandra's global character theory extend to this general setting. As an application we consider a general setup, for which we show that algebraic characters answer discretely decomposable branching problems.

math.RT↗
arXiv2013

Algebraic Characters of Harish-Chandra Modules and Arithmeticity

These are expanded notes from lectures at the Workshop "Representation Theory and Applications" held at Yeditepe University, Istanbul, in honor of Roger E. Howe. They are supplemented by the application of algebraic character theory to the construction of Galois-equivariant characters for Harish-Chandra modules.

math.NT↗
arXiv2009

Modular symbols for reductive groups and p-adic Rankin-Selberg convolutions over number fields

We give a construction of a wide class of modular symbols attached to reductive groups. As an application we construct a p-adic distribution interpolating the special values of the twisted Rankin-Selberg L-function attached to cuspidal automorphic representations of GL(n) and GL(n-1) over number fields. If the representations are ordinary at p, our distribution is bounded and yields analyticity of the associated p-adic L-function.

math.NT↗