Annihilator varieties of distinguished modules of reductive Lie algebras

We provide a microlocal necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs. Let ${\mathbf {G}}$ be a complex algebraic reductive group and ${\mathbf {H}}\subset {\mathbf {G}}$ be a spherical algebraic subgroup. Let ${\mathfrak {g}},{\mathfrak {h}}$ denote the Lie algebras of ${\mathbf {G}}$ and ${\mathbf {H}}$ , and let ${\mathfrak {h}}^{\bot }$ denote the orthogonal complement to ${\mathfrak {h}}$ in ${\mathfrak {g}}^*$ . A ${\mathfrak {g}}$ -module is called ${\mathfrak {h}}$ -distinguished if it admits a nonzero ${\mathfrak {h}}$ -invariant functional. We show that the maximal ${\mathbf {G}}$ -orbit in the annihilator variety of any irreducible ${\mathfrak {h}}$ -distinguished ${\mathfrak {g}}$ -module intersects ${\mathfrak {h}}^{\bot }$ . This generalises a result of Vogan [Vog91]. We apply this to Casselman–Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet modules. Further, we prove in many cases that – as suggested by [Pra19, Question 1] – when H is a symmetric subgroup of a real reductive group G, the existence of a tempered H-distinguished representation of G implies the existence of a generic H-distinguished representation of G. Many of the models studied in the theory of automorphic forms involve an additive character on the unipotent radical of the subgroup $\bf H$ , and we have devised a twisted version of our theorem that yields necessary conditions for the existence of those mixed models. Our method of proof here is inspired by the theory of modules over W-algebras. As an application of our theorem we derive necessary conditions for the existence of Rankin–Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan–Gross–Prasad conjectures for nongeneric representations [GGP20]. Finally, we provide more general results that ease the sphericity assumption on the subgroups, and apply them to local theta correspondence in type II and to degenerate Whittaker models.

“None’s Reflex”: Enactivism and Observational Philosophy on Consciousness and Observation

The paper is dedicated to the reconstruction of Alexander Piatigorsky’s observational philosophy within the context of the confrontation between two versions of the transcendental project of man-in-the-world. The first project accentuates the invariant functional organization of cognitive systems by abstracting from bodily, affective and phenomenological realization of this organization. On the contrary, the second project emphasizes the phenomenological perspective of the experience of givenness, always already dependent on whose experience this is and how the cognitive system living this experience is organized. The first project can be called functionalist, and the second – phenomenological. Ontological and epistemological positions of these projects are specified in the problem of the observer, its status in the world and cognitive practice. The observational philosophy possesses an intermediate position between these two programs since, aiming to disclose the invariant structure of observation, it proceeds from the factual experience of the embodied subject placed into the situation of self-observation and observation of the other subject. It is concluded that Piatigorsky’s philosophy borrows from the functionalist project the commitment to self-objectivation (observation of thinking is always the observation of the other thinking) and rejection from the spatiotemporal localization of cognitive activity (thinking is always “none’s” and does not belong to any kind of individual). With the phenomenological project of enactivism Piatigorsky shares the aspiration to disclose the invariant cognitive structures during the empirical observation of the real enactment of cognitive agency (the organization of cognitive systems is the same while its structural realizations are multiple), abandonment of substantialization of the self (“none’s” thinking is considered as the emergent effect of interaction among two or several observers – the autopoietic systems) as well as the refusal from theoretical formulation of the problem of consciousness (observational philosophy develops metatheoretical prolegomena to theory of consciousness, which in turn is considered as lived and essentially practical in phenomenology).

CR-HARMONIC MAPS

We develop the notion of renormalized energy in Cauchy–Riemann (CR) geometry for maps from a strictly pseudoconvex pseudo-Hermitian manifold to a Riemannian manifold. This energy is a CR invariant functional whose critical points, which we call CR-harmonic maps, satisfy a CR covariant partial differential equation. The corresponding operator coincides on functions with the CR Paneitz operator.

The strong Fatou property of risk measures

In this paper, we explore several Fatou-type properties of risk measures. The paper continues to reveal that the strong Fatou property,whichwas introduced in [19], seems to be most suitable to ensure nice dual representations of risk measures. Our main result asserts that every quasiconvex law-invariant functional on a rearrangement invariant space X with the strong Fatou property is (X, L1) lower semicontinuous and that the converse is true on a wide range of rearrangement invariant spaces. We also study inf-convolutions of law-invariant or surplus-invariant risk measures that preserve the (strong) Fatou property.

Control of diverse subcellular processes by a single multi-functional lipid phosphatidylinositol 4,5-bisphosphate [PI(4,5)P2]

Phosphatidylinositol 4,5-bisphosphate [PI(4,5)P2] is a multi-functional lipid that regulates several essential subcellular processes in eukaryotic cells. In addition to its well-established function as a substrate for receptor-activated signalling at the plasma membrane (PM), it is now recognized that distinct PI(4,5)P2 pools are present at other organelle membranes. However, a long-standing question that remains unresolved is the mechanism by which a single lipid species, with an invariant functional head group, delivers numerous functions without loss of fidelity. In the present review, we summarize studies that have examined the molecular processes that shape the repertoire of PI(4,5)P2 pools in diverse eukaryotes. Collectively, these studies indicate a conserved role for lipid kinase isoforms in generating functionally distinct pools of PI(4,5)P2 in diverse metazoan species. The sophistication underlying the regulation of multiple functions by PI(4,5)P2 is also shaped by mechanisms that regulate its availability to enzymes involved in its metabolism as well as molecular processes that control its diffusion at nanoscales in the PM. Collectively, these mechanisms ensure the specificity of PI(4,5)P2 mediated signalling at eukaryotic membranes.

The atypical cation-conduction and gating properties of ELIC underscore the marked functional versatility of the pentameric ligand-gated ion-channel fold

The superfamily of pentameric ligand-gated ion channels (pLGICs) is unique among ionotropic receptors in that the same overall structure has evolved to generate multiple members with different combinations of agonist specificities and permeant-ion charge selectivities. However, aside from these differences, pLGICs have been typically regarded as having several invariant functional properties. These include pore blockade by extracellular quaternary-ammonium cations in the micromolar-to-millimolar concentration range (in the case of the cation-selective members), and a gain-of-function phenotype, which manifests as a slower deactivation time course, as a result of mutations that reduce the hydrophobicity of the transmembrane pore lining. Here, we tested this notion on three distantly related cation-selective members of the pLGIC superfamily: the mouse muscle nicotinic acetylcholine receptor (nAChR), and the bacterial GLIC and ELIC channels. Remarkably, we found that, whereas low millimolar concentrations of TMA+ and TEA+ block the nAChR and GLIC, neither of these two quaternary-ammonium cations blocks ELIC at such concentrations; instead, both carry measurable inward currents when present as the only cations on the extracellular side. Also, we found that, whereas lidocaine binding speeds up the current-decay time courses of the nAChR and GLIC in the presence of saturating concentrations of agonists, the binding of lidocaine to ELIC slows this time course down. Furthermore, whereas mutations that reduce the hydrophobicity of the side chains at position 9′ of the M2 α-helices greatly slowed the deactivation time course of the nAChR and GLIC, these mutations had little effect—or even sped up deactivation—when engineered in ELIC. Our data indicate that caution should be exercised when generalizing results obtained with ELIC to the rest of the pLGICs, but more intriguingly, they hint at the possibility that ELIC is a representative of a novel branch of the superfamily with markedly divergent pore properties despite a well-conserved three-dimensional architecture.

Specification Issues in Assessing the Moderating Role of Issue Importance: A Comment on Grynaviski and Corrigan (2006)

The empirical study of spatial issue voting has experienced a resurgence in recent years due to advances in data collection and research design. Grynaviski and Corrigan make several important contributions to this literature. In this note, we comment on one of Grynaviski and Corrigan's recommendations—to not include a main effect for issue importance when estimating models assessing the interactive relationship between importance and policy proximity. According to the authors, including the main importance term is incorrect because it is not necessary in representing a scale-invariant functional form under some assumptions and is insufficient under others. In deriving their reduced-form expression, the authors produce a model that is unintuitive and inappropriate for most data. Moreover, the restrictions Grynaviski and Corrigan impose on their model can produce perverse empirical predictions. We show that a model including main effect terms for importance is indeed scale invariant and that inclusion of the main importance term is necessary for scale invariance with respect to partial utility functions.