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Plantar fasciitis is one of the most common pathologies addressed by foot and ankle surgeons. Despite advances and overall success rates for conservative therapy, many of the recalcitrant cases proceed to require surgical correction. Partial to complete release of the fascia is often performed altering foot biomechanics and severing the windlass mechanism. Endoscopic debridement of the plantar fascia allows for direct visualization and removal of the inflammatory tissue while leaving the fascia and its function intact. A total of 125 feet were evaluated with a minimum follow-up time of 5 years. Gender, body mass index, and duration of symptoms were all evaluated and documented. Visual analog scale scores (VAS), American Orthopedic Foot and Ankle Score (AOFAS), and Foot Function Index (FFI) were collected both pre- and postoperatively. AOFAS, FFI, and VAS scores improved from a pre-operative mean of 57.6, 89.4%, and 8.6-89.1, 13.4%, and 0.7 respectively (p.05) at final follow-up. Of the 125 patients, 98% stated they were satisfied with the operative outcome and would undergo the procedure again. At final follow-up, no patient suffered rupture of the fascia or recurrence. Patients were able to bear weight immediately following the surgery in a walking boot and on average patients were able to return to work at 3.4 days following surgery. This is a novel technique that does not compromise the plantar fascia or alter foot biomechanics with promising 5-year outcomes.

This study of 1179 undergraduates with depression examined whether students revealed their depression to their online science instructors and the reasoning behind their decisions. Few students revealed their depression, but perceived that if they did, they would benefit by receiving accommodations but risk instructor judgment.

On the Ground • The 2020 SRM Annual Meeting piloted “Campfire Conversation,” round-table discussions styled after the World Cafe approach. • The event attracted 280 attendees and enabled multidirectional knowledge exchange (i.e., “cuss and discuss”), rather than one-way “chalk-and-talk.” Attendees participated in three 20-minute facilitated round-table discussions around three topics they selected from a menu of 13 timely rangeland issues. • Change was a common theme for many Campfire Conversations, including social, climatic, ecological, management, and policy changes. • Participants highlighted a desire for SRM to grow as an organization by enhancing members’ opportunities and resources for multidirectional knowledge exchange among students, scientists, practitioners, and policy-makers; cross-generational mentorship; cross-disciplinary training; diverse ways of knowing; and greater inclusivity and connection to the SRM. • A post-event analysis of the Campfire Conversation event revealed valuable lessons for organizing successful World Cafe-style sessions at future SRM meetings, including virtual meetings.

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here we carry out a major step in the structural analysis of geodesic orbit Lorentz nilmanifolds. Those are the geodesic orbit Lorentz manifolds $M = G/H$ such that a nilpotent analytic subgroup of $G$ is transitive on $M$. Suppose that there is a reductive decomposition $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{n}$ (vector space direct sum) with $\mathfrak{n}$ nilpotent. When the metric is nondegenerate on $[\mathfrak{n},\mathfrak{n}]$ we show that $\mathfrak{n}$ is abelian or 2-step nilpotent (this is the same result as for geodesic orbit Riemannian nilmanifolds), and when the metric is degenerate on $[\mathfrak{n},\mathfrak{n}]$ we show that $\mathfrak{n}$ is a Lorentz double extension corresponding to a geodesic orbit Riemannian nilmanifold. In the latter case we construct examples to show that the number of nilpotency steps is unbounded.

Two homogeneous pseudo-riemannian manifolds $(G/H, ds^2)$ and $(G'/H', ds'^2)$ belong to the same {\it real form family} if their complexifications $(G_{\mathbb C}/H_{\mathbb C}, ds_{\mathbb C}^2)$ and $(G'_{\mathbb C}/H'_{\mathbb C}, ds'^2_{\mathbb C})$ are isometric. The point is that in many cases a particular space $(G/H, ds^2)$ has interesting properties, and those properties hold for the spaces in its real form family. Here we prove that if $(G/H, ds^2)$ is a geodesic orbit space with a reductive decomposition $\mathfrak{g} = \mathfrak{h} + \mathfrak{m}$, then the same holds all the members of its real form family. In particular our understanding of compact geodesic orbit riemannian manifolds gives information on geodesic orbit pseudo-riemannian manifolds. We also prove similar results for naturally reductive spaces, for commutative spaces, and in most cases for weakly symmetric spaces. We end with a discussion of inclusions of these real form families, a discussion of D'Atri spaces, and a number of open problems., Comment: This note complements earlier work of Zhiqi Chen and the author

Anxiety is the top mental health concern for undergraduates. While researchers have identified ways that in-person science courses can affect anxiety, little is known about how online science courses affect anxiety. In this study, 2111 undergraduates at a large research-intensive institution completed survey questions about their anxiety in large-enrollment online science courses. Specifically, we assessed students’ anxiety in the context of online science courses and asked what aspects of online science courses increase and decrease their anxiety. Students also identified what instructors can do to lessen anxiety in online classrooms. We used open coding and logistic regression to analyze student responses. More than 50% of students reported at least moderate anxiety in the context of online college science courses. Students commonly reported that the potential for personal technology issues (69.8%) and proctored exams (68.0%) increased their anxiety, while being able to access content at a later time (79.0%) and attending class from where they want (74.2%) decreased their anxiety. The most common ways that students suggested that instructors could decrease student anxiety is to increase test-taking flexibility (25.0%) and be understanding (23.1%). This study provides insight into how instructors can create more inclusive online learning environments for students with anxiety.

ObjectiveThe area postrema (AP) and the nucleus tractus solitaris (NTS), located in the hindbrain, are key nuclei that sense and integrate peripheral nutritional signals and, consequently, regulate feeding behaviour. While single cell transcriptomics have been used in mice to reveal the gene expression profile and heterogeneity of key hypothalamic populations, similar in-depth studies have not yet been performed in the hindbrain.MethodsUsing single-nucleus RNA sequencing, we provide a detailed survey of 16,034 cells within the AP and NTS of the mouse, in the fed and fasted state.ResultsOf these, 8910 are neurons that group into 30 clusters, with 4289 coming from mice fedad libitumand 4621 from overnight fasted mice. 7124 nuclei are from non-neuronal cells, including oligodendrocytes, astrocytes and microglia. Interestingly, we identified that the oligodendrocyte population was particularly transcriptionally sensitive to an overnight fast. The receptors GLP1R, GIPR, GFRAL and CALCR, which bind GLP1, GIP, GDF15 and amylin respectively, are all expressed in the hindbrain and are major targets for anti-obesity therapeutics. We characterise the transcriptomes of these four populations and show that their gene expression profiles are not dramatically altered by an overnight fast. Notably, we find that roughly half of cells that express GIPR are oligodendrocytes. Additionally, we profile POMC expressing neurons within the hindbrain and demonstrate that 84% of POMC neurons express either PCSK1, PSCK2 or both, implying that melanocortin peptides are likely produced by these neurons.ConclusionWe provide a detailed single-cell level characterisation of AP and NTS cells expressing receptors for key anti-obesity drugs that are either already approved for human use or are in clinical trials. This resource will help delineate the mechanisms underlying the effectiveness of these compounds, and also prove useful in the continued search for other novel therapeutic targets.

In this note we study globally homogeneous Riemannian quotients $\Gamma\backslash (M,ds^2)$ of homogeneous Riemannian manifolds $(M,ds^2)$. The Homogeneity Conjecture is that $\Gamma\backslash (M,ds^2)$ is (globally) homogeneous if and only if $(M,ds^2)$ is homogeneous and every $\gamma \in \Gamma$ is of constant displacement on $(M,ds^2)$. We provide further evidence for that conjecture by (i) verifying it for normal homogeneous Riemannian manifolds that also admit an invariant Riemannian metric of strictly positive sectional curvature and (ii) showing that in most (three or less) cases the normality condition can be dropped.

Simple Summary On weekends, people do things differently from weekdays, such as dining at a restaurant, going to a night club, attending a concert or a sporting event, or simply staying up late. These leisure activities in the city can change the environment people live in and can hurt wildlife that also lives in the same city. We recorded bats in the city center and in the city periphery and compared how active bats were. We found that in the city center, bats were less active on weekends than weekdays. The opposite pattern was found in the city periphery. It is possible that bats moved from the city center to the city periphery on weekends. Thus, continuous greenways are important to facilitate bat movements and avoid human–wildlife conflict. City planners can add new parks and/or preserve old-growth vegetation to form the center-to-periphery greenways. Abstract In the urban environment, wildlife faces novel human disturbances in unique temporal patterns. The weekend effect describes that human activities on weekends trigger changes in the environment and impact wildlife negatively. Reduced occurrence, altered behaviors, and/or reduced fitness have been found in birds, ungulates, and meso-carnivores due to the weekend effect. We aimed to investigate if urban bat activity would differ on weekends from weekdays. We analyzed year-round bat acoustic monitoring data collected from two sites near the city center and two sites in the residential area/park complex in the city periphery. We constructed generalized linear models and found that bat activity was significantly lower on weekends as compared to weekdays during spring and summer at the site in the open space near the city center. In contrast, during the same seasons, the sites in the city periphery showed increased bat activity on weekends. Hourly bat activity overnight suggested that bats might move from the city center to the periphery on weekends. We demonstrated the behavioral adaptability in urban wildlife for co-existing with human. We recommend that urban planning should implement practices such as adding new greenspaces and/or preserving old-growth vegetation to form continuous greenways from the city center to the city periphery as corridors to facilitate bat movements and reduce possible human-wildlife conflict.

Visual Abstract, Highlights • The GLP-1RAs liraglutide and semaglutide reduce cardiovascular risk in type 2 diabetes patients. • In ApoE−/− mice and LDLr−/− mice, liraglutide and semaglutide treatment significantly attenuated plaque lesion development, in part independently of body weight and cholesterol lowering. • Semaglutide decreased levels of plasma markers of systemic inflammation in an acute inflammation model (lipopolysaccharide), and transcriptomic analysis of aortic atherosclerotic tissue revealed that multiple inflammatory pathways were down-regulated by semaglutide., Summary The glucagon-like peptide-1 receptor agonists (GLP-1RAs) liraglutide and semaglutide reduce cardiovascular risk in type 2 diabetes patients. The mode of action is suggested to occur through modified atherosclerotic progression. In this study, both of the compounds significantly attenuated plaque lesion development in apolipoprotein E-deficient (ApoE−/−) mice and low-density lipoprotein receptor-deficient (LDLr−/−) mice. This attenuation was partly independent of weight and cholesterol lowering. In aortic tissue, exposure to a Western diet alters expression of genes in pathways relevant to the pathogenesis of atherosclerosis, including leukocyte recruitment, leukocyte rolling, adhesion/extravasation, cholesterol metabolism, lipid-mediated signaling, extracellular matrix protein turnover, and plaque hemorrhage. Treatment with semaglutide significantly reversed these changes. These data suggest GLP-1RAs affect atherosclerosis through an anti-inflammatory mechanism.

We develop the classification of weakly symmetric pseudo--riemannian manifolds $G/H$ where $G$ is a semisimple Lie group and $H$ is a reductive subgroup. We derive the classification from the cases where $G$ is compact, and then we discuss the (isotropy) representation of $H$ on the tangent space of $G/H$ and the signature of the invariant pseudo--riemannian metric. As a consequence we obtain the classification of semisimple weakly symmetric manifolds of Lorentz signature $(n-1,1)$ and trans--Lorentz (conformal Lorentz) signature $(n-2,2)$., Comment: This arXiv version 3 adds a reference to version 2 and tweaks the Introduction accordingly

Objective The area postrema (AP) and nucleus tractus solitarius (NTS) located in the hindbrain are key nuclei that sense and integrate peripheral nutritional signals and consequently regulate feeding behaviour. While single-cell transcriptomics have been used in mice to reveal the gene expression profile and heterogeneity of key hypothalamic populations, similar in-depth studies have not yet been performed in the hindbrain. Methods Using single-nucleus RNA sequencing, we provide a detailed survey of 16,034 cells within the AP and NTS of mice in the fed and fasted states. Results Of these, 8,910 were neurons that group into 30 clusters, with 4,289 from mice fed ad libitum and 4,621 from overnight fasted mice. A total of 7,124 nuclei were from non-neuronal cells, including oligodendrocytes, astrocytes, and microglia. Interestingly, we identified that the oligodendrocyte population was particularly transcriptionally sensitive to an overnight fast. The receptors GLP1R, GIPR, GFRAL, and CALCR, which bind GLP1, GIP, GDF15, and amylin, respectively, are all expressed in the hindbrain and are major targets for anti-obesity therapeutics. We characterise the transcriptomes of these four populations and show that their gene expression profiles are not dramatically altered by an overnight fast. Notably, we find that roughly half of cells that express GIPR are oligodendrocytes. Additionally, we profile POMC-expressing neurons within the hindbrain and demonstrate that 84% of POMC neurons express either PCSK1, PSCK2, or both, implying that melanocortin peptides are likely produced by these neurons. Conclusion We provide a detailed single-cell level characterisation of AP and NTS cells expressing receptors for key anti-obesity drugs that are either already approved for human use or in clinical trials. This resource will help delineate the mechanisms underlying the effectiveness of these compounds and also prove useful in the continued search for other novel therapeutic targets., Highlights • We provide a survey of 16,034 cells of the murine AP and NTS in the fed and fasted state by single-nucleus RNA sequencing. • 8,910 are grouped into 30 neuronal clusters, and 7,124 are non-neuronal such as oligodendrocytes, astrocytes and microglia. • We examine the transcriptomes of populations expressing Glp1r, Gipr, Gfral, Calcr, and Pomc. • The expression profiles for the above populations were not dramatically altered by an overnight fast. • We provide a detailed single-cell level characterisation of AP and NTS cells expressing anti-obesity drug targets.

Semaglutide, a glucagon-like peptide 1 (GLP-1) analog, induces weight loss, lowers glucose levels, and reduces cardiovascular risk in patients with diabetes. Mechanistic preclinical studies suggest weight loss is mediated through GLP-1 receptors (GLP-1Rs) in the brain. The findings presented here show that semaglutide modulated food preference, reduced food intake, and caused weight loss without decreasing energy expenditure. Semaglutide directly accessed the brainstem, septal nucleus, and hypothalamus but did not cross the blood-brain barrier; it interacted with the brain through the circumventricular organs and several select sites adjacent to the ventricles. Semaglutide induced central c-Fos activation in 10 brain areas, including hindbrain areas directly targeted by semaglutide, and secondary areas without direct GLP-1R interaction, such as the lateral parabrachial nucleus. Automated analysis of semaglutide access, c-Fos activity, GLP-1R distribution, and brain connectivity revealed that activation may involve meal termination controlled by neurons in the lateral parabrachial nucleus. Transcriptomic analysis of microdissected brain areas from semaglutide-treated rats showed upregulation of prolactin-releasing hormone and tyrosine hydroxylase in the area postrema. We suggest semaglutide lowers body weight by direct interaction with diverse GLP-1R populations and by directly and indirectly affecting the activity of neural pathways involved in food intake, reward, and energy expenditure.

In this note we complete a study of globally homogeneous Riemannian quotients $\Gamma\backslash (M,ds^2)$ in positive curvature. Specifically, $M$ is a homogeneous space $G/H$ that admits a $G$-invariant Riemannian metric of strictly positive sectional curvature, and $ds^2$ is a $G$--invariant Riemannian metric on $M$, not necessarily normal and not necessarily positively curved. The Homogeneity Conjecture is that $\Gamma\backslash (M,ds^2)$ is (globally) homogeneous if and only if $(M,ds^2)$ is homogeneous and every $\gamma \in \Gamma$ is of constant displacement on $(M,ds^2)$. In an earlier paper we verified that conjecture for all homogeneous spaces that admit an invariant Riemannian metric of positive curvature -- with three exceptions, all odd dimensional spheres, which surprisingly did not yield to the earlier approaches. Here we develop some methods that let us verify the Homogeneity Conjecture for those three obstinate spheres. That completes verification of the Homogeneity Conjecture in positive curvature., Comment: 10 pages

© 2018 Académie des sciences In this paper, we investigate left-invariant geodesic orbit metrics on connected simple Lie groups, where the metrics are formed by the structures of flag manifolds. We prove that all these left-invariant geodesic orbit metrics on simple Lie groups are naturally reductive.

Author(s): Wolf, JA | Abstract: This is a semi-expository update and rewrite of my 1974 AMS AMS Memoir describing Plancherel formulae and partial Dolbeault cohomology realizations for standard tempered representations for general real reductive Lie groups. Even after so many years, much of that Memoir is up to date, but of course there have been a number of refinements, advances and new developments, most of which have applied to smaller classes of real reductive Lie groups. Here we rewrite that AMS Memoir in in view of these advances and indicate the ties with some of the more recent (or at least less classical) approaches to geometric realization of unitary representations.

A compact Riemannian homogeneous space G/H, with a bi-invariant orthogonal decomposition g=h+m is called positively curved for commuting pairs, if the sectional curvature vanishes for any tangent plane in TeH(G/H) spanned by a linearly independent commuting pair in m. In this paper, we will prove that on the coset space Sp(2)/U(1), in which U(1) corresponds to a short root, admits positively curved metrics for commuting pairs. B. Wilking recently proved that this Sp(2)/U(1) cannot be positively curved in the general sense. This is the first example to distinguish the set of compact coset spaces admitting positively curved metrics, and that for metrics positively curved only for commuting pairs.

An intergenic region of human chromosome 2 (2p25.3) harbors genetic variants which are among those most strongly and reproducibly associated with obesity. The gene closest to these variants is TMEM18, although the molecular mechanisms mediating these effects remain entirely unknown. Tmem18 expression in the murine hypothalamic paraventricular nucleus (PVN) was altered by changes in nutritional state. Germline loss of Tmem18 in mice resulted in increased body weight, which was exacerbated by high fat diet and driven by increased food intake. Selective overexpression of Tmem18 in the PVN of wild-type mice reduced food intake and also increased energy expenditure. We provide evidence that TMEM18 has four, not three, transmembrane domains and that it physically interacts with key components of the nuclear pore complex. Our data support the hypothesis that TMEM18 itself, acting within the central nervous system, is a plausible mediator of the impact of adjacent genetic variation on human adiposity.

Objective Arcuate proopiomelanocortin (POMC) neurons are critical nodes in the control of body weight. Often characterized simply as direct targets for leptin, recent data suggest a more complex architecture. Methods Using single cell RNA sequencing, we have generated an atlas of gene expression in murine POMC neurons. Results Of 163 neurons, 118 expressed high levels of Pomc with little/no Agrp expression and were considered “canonical” POMC neurons (P+). The other 45/163 expressed low levels of Pomc and high levels of Agrp (A+P+). Unbiased clustering analysis of P+ neurons revealed four different classes, each with distinct cell surface receptor gene expression profiles. Further, only 12% (14/118) of P+ neurons expressed the leptin receptor (Lepr) compared with 58% (26/45) of A+P+ neurons. In contrast, the insulin receptor (Insr) was expressed at similar frequency on P+ and A+P+ neurons (64% and 55%, respectively). Conclusion These data reveal arcuate POMC neurons to be a highly heterogeneous population. Accession Numbers: GSE92707., Highlights • Hypothalamic POMC neurons are heterogeneous and can broadly divided into 4 groups. • Cell surface receptors are major drivers for the segregation. • Unexpectedly, 28% of POMC-cells show signatures typical of AgRP/NPY neurons. • Only 12% express leptin receptor, indicating response to leptin is likely indirect.

An intergenic region of human Chromosome 2 (2p25.3) harbours genetic variants which are among those most strongly and reproducibly associated with obesity. The molecular mechanisms mediating these effects remain entirely unknown. The gene closest to these variants is TMEM18, encoding a transmembrane protein localised to the nuclear membrane. The expression of Tmem18 within the murine hypothalamic paraventricular nucleus was altered by changes in nutritional state, with no significant change seen in three other closest genes. Germline loss of Tmem18 in mice resulted in increased body weight, which was exacerbated by high fat diet and driven by increased food intake. Selective overexpression of Tmem18 in the PVN of wild-type mice reduced food intake and also increased energy expenditure. We confirmed the nuclear membrane localisation of TMEM18 but provide new evidence that it is has four, not three, transmembrane domains and that it physically interacts with key components of the nuclear pore complex. Our data support the hypothesis that TMEM18 itself, acting within the central nervous system, is a plausible mediator of the impact of adjacent genetic variation on human adiposity.

In this paper we develop the basic tools for a classification of Killing vector fields of constant length on pseudo-Riemannian homogeneous spaces. This extends a recent paper of M. Xu and J. A. Wolf, which classified the pairs $(M,\xi)$ where $M = G/H$ is a Riemannian normal homogeneous space, G is a compact simple Lie group, and $\xi \in \mathfrak{g}$ defines a nonzero Killing vector field of constant length on $M$. The method there was direct computation. Here we make use of the moment map $M \to \mathfrak{g}^{*}$ and the flag manifold structure of $\mathrm{Ad} (G) \xi$ to give a shorter, more geometric proof which does not require compactness and which is valid in the pseudo-Riemannian setting. In that context we break the classification problem into three parts. The first is easily settled. The second concerns the cases where $\xi$ is elliptic and $G$ is simple (but not necessarily compact); that case is our main result here. The third, which remains open, is a more combinatorial problem involving elements of the first two.

In this paper we give an explicit description of the bounded displacement isometries of a class of spaces that includes the Riemannian nilmanifolds. The class of spaces consists of metric spaces (and thus includes Finsler manifolds) on which an exponential solvable Lie group acts transitively by isometries. The bounded isometries are proved to be of constant displacement. Their characterization gives further evidence for the author's 1962 conjecture on homogeneous Riemannian quotient manifolds. That conjecture suggests that if $\Gamma \backslash M$ is a Riemannian quotient of a connected simply connected homogeneous Riemannian manifold $M$, then $\Gamma \backslash M$ is homogeneous if and only if each isometry $\gamma \in \Gamma$ is of constant displacement. Our description of bounded isometries gives an alternative proof of an old result of J. Tits on bounded automorphisms of semisimple Lie groups. The topic of constant displacement isometries has an interesting history, starting with Clifford's use of quaternions in non--euclidean geometry, and we sketch that in a historical note., Comment: This version contains more background material than the first version, and the organization of the paper has been modified. The mathematical content is the same

Summary Ambiguity regarding the role of glucose-dependent insulinotropic polypeptide (GIP) in obesity arises from conflicting reports asserting that both GIP receptor (GIPR) agonism and antagonism are effective strategies for inhibiting weight gain. To enable identification and manipulation of Gipr-expressing (Gipr) cells, we created Gipr-Cre knockin mice. As GIPR-agonists have recently been reported to suppress food intake, we aimed to identify central mediators of this effect. Gipr cells were identified in the arcuate, dorsomedial, and paraventricular nuclei of the hypothalamus, as confirmed by RNAscope in mouse and human. Single-cell RNA-seq identified clusters of hypothalamic Gipr cells exhibiting transcriptomic signatures for vascular, glial, and neuronal cells, the latter expressing somatostatin but little pro-opiomelanocortin or agouti-related peptide. Activation of Gq-DREADDs in hypothalamic Gipr cells suppressed food intake in vivo, which was not obviously additive with concomitant GLP1R activation. These data identify hypothalamic GIPR as a target for the regulation of energy balance., Graphical Abstract, Highlights • Gipr is expressed in the hypothalamus, as demonstrated using a new Gipr-Cre mouse model • Gipr cells included somatostatin-positive neurons, glia, and vascular cells • Gipr overlapped partially with Glp1r in human and mouse hypothalamus by RNAscope • Activation of Gipr neurons using local AAV-delivered Gq-DREADDs reduced food intake, Adriaenssens et al. identified cells expressing receptors for the gut hormone GIP in human and mouse hypothalamus using a new Gipr-Cre mouse model and RNAscope. Local hypothalamic delivery of AAVs expressing Cre-dependent Gq-DREADDs revealed that activation of Gipr-positive neurons in mice reduced food consumption.

We extend the Paley–Wiener theorem for Riemannian symmetric spaces to an important class of infinite-dimensional symmetric spaces. For this we define a notion of propagation of symmetric spaces and examine the direct (injective) limit symmetric spaces defined by propagation. This relies on some of our earlier work on invariant differential operators and the action of Weyl group invariant polynomials under restriction.

We study the conditions for a nilpotent Lie group to be foliated into subgroups that have square integrable (relative discrete series) unitary representations, that fit together to form a filtration by normal subgroups. Then we use that filtration to construct a class of “stepwise square integrable” representations on which Plancherel measure is concentrated. Further, we work out the character formulae for those stepwise square integrable representations, and we give an explicit Plancherel formula. Next, we use some structure theory to check that all these constructions and results apply to nilradicals of minimal parabolic subgroups of real reductive Lie groups. Finally, we develop multiplicity formulae for compact quotients $$N/\varGamma $$ where $$\varGamma $$ respects the filtration.

Prader-Willi syndrome (PWS) is a complex disorder that manifests with an array of phenotypes, such as hypotonia and difficulties in feeding during infancy and reduced energy expenditure, hyperphagia, and developmental delays later in life. While the genetic cause has long been known, it is still not clear how mutations at this locus produce this array of phenotypes. In this issue of the JCI, Burnett and colleagues used a comprehensive approach to gain insight into how PWS-associated mutations drive disease. Using neurons derived from PWS patient induced pluripotent stem cells (iPSCs) and mouse models, the authors provide evidence that neuroendocrine PWS-associated phenotypes may be linked to reduced expression of prohormone convertase 1 (PC1). While these compelling results support a critical role for PC1 deficiency in PWS, more work needs to be done to fully understand how and to what extent loss of this prohormone processing enzyme underlies disease manifestations in PWS patients.

We study Riemannian coverings φ : M ˜ → Γ \ M ˜ where M ˜ is a normal homogeneous space G / K 1 fibered over another normal homogeneous space M = G / K and K is locally isomorphic to a nontrivial product K 1 × K 2 . The most familiar such fibrations π : M ˜ → M are the natural fibrations of Stiefel manifolds S O ( n 1 + n 2 ) / S O ( n 1 ) over Grassmann manifolds S O ( n 1 + n 2 ) / [ S O ( n 1 ) × S O ( n 2 ) ] and the twistor space bundles over quaternionic symmetric spaces (= quaternion-Kaehler symmetric spaces = Wolf spaces). The most familiar of these coverings φ : M ˜ → Γ \ M ˜ are the universal Riemannian coverings of spherical space forms. When M = G / K is reasonably well understood, in particular when G / K is a Riemannian symmetric space or when K is a connected subgroup of maximal rank in G, we show that the Homogeneity Conjecture holds for M ˜ . In other words we show that Γ \ M ˜ is homogeneous if and only if every γ ∈ Γ is an isometry of constant displacement. In order to find all the isometries of constant displacement on M ˜ we work out the full isometry group of M ˜ , extending Elie Cartan's determination of the full group of isometries of a Riemannian symmetric space. We also discuss some pseudo-Riemannian extensions of our results.

© 2016, Springer Science+Business Media New York. Killing vector fields of constant length correspond to isometries of constant displacement. Those in turn have been used to study homogeneity of Riemannian and Finsler quotient manifolds. Almost all of that work has been done for group manifolds or, more generally, for symmetric spaces. This paper extends the scope of research on constant length Killing vector fields to a class of Riemannian normal homogeneous spaces.

There are some new developments on Plancherel formula and growth of matrix coefficients for unitary representations of nilpotent Lie groups. These have several consequences for the geometry of weakly symmetric spaces and analysis on parabolic subgroups of real semisimple Lie groups, and to (infinite dimensional) locally nilpotent Lie groups. Many of these consequences are still under development. In this note I’ll survey a few of these new aspects of representation theory for nilpotent Lie groups and parabolic subgroups.

We examine the structure of the Levi component MA in a minimal parabolic subgroup $$P = MAN$$ of a real reductive Lie group G and work out the cases where M is metabelian, equivalently where $$\mathfrak {p}$$ is solvable. When G is a linear group we verify that $$\mathfrak {p}$$ is solvable if and only if M is commutative. In the general case M is abelian modulo the center $$Z_G$$ , we indicate the exact structure of M and P, and we work out the precise Plancherel Theorem and Fourier Inversion Formulae. This lays the groundwork for comparing tempered representations of G with those induced from generic representations of P.

We consider the complex ind-group \(G=\mathrm {SL}(\infty ,\mathbb {C})\) and its real forms \(G^0={\mathrm {SU}}(\infty ,\infty )\), \({\mathrm {SU}}(p,\infty )\), \(\mathrm {SL}(\infty ,\mathbb {R})\), \(\mathrm {SL}(\infty ,\mathbb {H})\). Our main object of study are the \(G^0\)-orbits on an ind-variety G / P for an arbitrary splitting parabolic ind-subgroup \(P\subset G\), under the assumption that the subgroups \(G^0\subset G\) and \(P\subset G\) are aligned in a natural way. We prove that the intersection of any \(G^0\)-orbit on G / P with a finite-dimensional flag variety \(G_n/P_n\) from a given exhaustion of G / P via \(G_n/P_n\) for \(n\rightarrow \infty \), is a single \((G^0\cap G_n)\)-orbit. We also characterize all ind-varieties G / P on which there are finitely many \(G^0\)-orbits, and provide criteria for the existence of open and closed \(G^0\)-orbits on G / P in the case of infinitely many \(G^0\)-orbits.

MGE: Research supported by the Australian Research Council. JAW: Research partially supported by NSF Grant DMS 99-88643.

In the study of complex flag manifolds, flag domains and their cycle spaces, a key point is the fact that the cycle space $\mathcal M_D$ of a flag domain $D$ is a Stein manifold. That fact has a long history. The earliest approach relied on construction of a strictly plurisubharmonic function on $\mathcal M_D$, starting with a $q$--convex exhaustion function on $D$, where $q$ is the dimension of a particular maximal compact subvariety of $D$ (we use the normalization that 0--convex means Stein). Construction of that exhaustion function on $D$ required that $D$ be measurable. In that case the exhaustion on $D$ was transferred to $\mathcal M_D$, using a special case of a method due to Barlet. Here we do the opposite: we use an incidence method to construct a canonical plurisubharmonic exhaustion function on $\mathcal M_D$ and use it in turn to construct a canonical $q$--convex exhaustion function on $D$. This promises to have strong consequences for cohomology vanishing theorems and the construction of admissible representations of real reductive Lie groups., Comment: 9 pages

In earlier papers we studied direct limits $${(G,\,K) = \varinjlim\, (G_n,K_n)}$$ of two types of Gelfand pairs. The first type was that in which the G n /K n are compact Riemannian symmetric spaces. The second type was that in which $${G_n = N_n\rtimes K_n}$$ with N n nilpotent, in other words pairs (G n , K n ) for which G n /K n is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections $${\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)}$$ for m ≧ n and prove that the left regular representation of G on the Hilbert space direct limit $${L^2(G/K) := \varinjlim L^2(G_n/K_n)}$$ is multiplicity-free. This left open questions concerning the nature of the elements of L 2(G/K). Here we define spaces $${\mathcal{A}(G_n/K_n)}$$ of regular functions on G n /K n and injections $${\nu_{m,n} : \mathcal{A}(G_n/K_n) \to \mathcal{A}(G_m/K_m)}$$ for m ≧ n related to restriction by $${\nu_{m,n}(f)|_{G_n/K_n} = f}$$ . Thus the direct limit $${\mathcal{A}(G/K) := \varinjlim \{\mathcal{A}(G_n/K_n), \nu_{m,n}\}}$$ sits as a particular G-submodule of the much larger inverse limit $${\varprojlim \{\mathcal{A}(G_n/K_n), {\rm restriction}\}}$$ . Further, we define a pre Hilbert space structure on $${\mathcal{A}(G/K)}$$ derived from that of L 2(G/K). This allows an interpretation of L 2(G/K) as the Hilbert space completion of the concretely defined function space $${\mathcal{A}(G/K)}$$ , and also defines a G-invariant inner product on $${\mathcal{A}(G/K)}$$ for which the left regular representation of G is multiplicity-free.