Rankin–Selberg L-functions via good sections

In this article, we revisit Rankin–Selberg integrals established by Jacquet, Piatetski-Shapiro and Shalika. We prove the equality of Rankin–Selberg local factors defined with Schwartz–Bruhat functions and the factors attached to good sections, introduced by Piatetski-Shapiro and Rallis. Moreover, we propose a notion of exceptional poles in the framework of good sections. For cases of Rankin–Selberg, Asai and exterior square L-functions, the exceptional poles are consistent with well-known exceptional poles which characterize certain distinguished representations.

Multiplicity One for Pairs of Prasad–Takloo-Bighash Type

Let ${\textrm{E}}/{\textrm{F}}$ be a quadratic extension of non-archimedean local fields of characteristic different from $2$. Let ${\textrm{A}}$ be an ${\textrm{F}}$-central simple algebra of even dimension so that it contains ${\textrm{E}}$ as a subfield, set ${\textrm{G}}={\textrm{A}}^\times $ and ${\textrm{H}}$ for the centralizer of ${\textrm{E}}^\times $ in ${\textrm{G}}$. Using a Galois descent argument, we prove that all double cosets ${\textrm{H}} g {\textrm{H}}\subset{\textrm{G}}$ are stable under the anti-involution $g\mapsto g^{-1}$, reducing to Guo’s result for ${\textrm{F}}$-split ${\textrm{G}}$ [14], which we extend to fields of positive characteristic different from $2$. We then show, combining global and local results, that ${\textrm{H}}$-distinguished irreducible representations of ${\textrm{G}}$ are self-dual and this implies that $({\textrm{G}},{\textrm{H}})$ is a Gelfand pair $$\begin{equation*}\operatorname{dim}_{\mathbb{C}}(\operatorname{Hom}_{{\textrm{H}}}(\pi,\mathbb{C}))\leq 1\end{equation*}$$for all smooth irreducible representations $\pi $ of ${\textrm{G}}$. Finally we explain how to obtain the multiplicity one statement in the archimedean case using the criteria of Aizenbud and Gourevitch ([1]), and we then show self-duality of irreducible distinguished representations in the archimedean case too.

Live Your Myth in Greece: Towards the Construction of a Heritage Identity

Nowadays, top-rated tourist attractions in Greece are ancient archaeological places and islands with blue-and-white esthetics. The country’s projected impression is greatly based on these two distinguished representations, chosen for their distinctive architecture scattered in the Greek landscape. Both imageries seem to be officially promoted in order to configure today’s national identity. The classical antiquities are related to the birthplace of European civilization, whereas the notion of the unspoilt archipelago with the whitewashed Cycladic houses works as a symbol of purity and eternity. The present article focuses on the analysis of these two Greek heritage scenarios and, subsequently, on their deconstruction. It aims to investigate the interaction between myth and reality and their role in forming the perception of contemporary Greece. The article argues that there is not a unique architectural history to come to light and, therefore, the highlighting of specific periods of it probably conceals intentions concerning patrimony management: selective excavation among the layers of history, historic preservation of selected buildings, and laws which impose the maintenance of certain findings or specific colors are some indicative signs. It also investigates the ways in which national heritage is directed and affected according to certain policies—local or foreign—that aim at a cultural investment in the world history.

ON -DISTINGUISHED REPRESENTATIONS OF THE QUASI-SPLIT UNITARY GROUPS

We study $\text{Sp}_{2n}(F)$ -distinction for representations of the quasi-split unitary group $U_{2n}(E/F)$ in $2n$ variables with respect to a quadratic extension $E/F$ of $p$ -adic fields. A conjecture of Dijols and Prasad predicts that no tempered representation is distinguished. We verify this for a large family of representations in terms of the Mœglin–Tadić classification of the discrete series. We further study distinction for some families of non-tempered representations. In particular, we exhibit $L$ -packets with no distinguished members that transfer under base change to $\text{Sp}_{2n}(E)$ -distinguished representations of $\text{GL}_{2n}(E)$ .

Gamma Factors, Root Numbers, and Distinction

We study a relation between distinction and special values of local invariants for representations of the general linear group over a quadratic extension of p-adic fields. We show that the local Rankin–Selberg root number of any pair of distinguished representation is trivial, and as a corollary we obtain an analogue for the global root number of any pair of distinguished cuspidal representations. We further study the extent to which the gamma factor at 1/2 is trivial for distinguished representations as well as the converse problem.