The infinitesimal characters of discrete series for real spherical spaces

Let $Z=G/H$ be the homogeneous space of a real reductive group and a unimodular real spherical subgroup, and consider the regular representation of $G$ on $L^2(Z)$. It is shown that all representations of the discrete series, that is, the irreducible subrepresentations of $L^2(Z)$, have infinitesimal characters which are real and belong to a lattice. Moreover, let $K$ be a maximal compact subgroup of $G$. Then each irreducible representation of $K$ occurs in a finite set of such discrete series representations only. Similar results are obtained for the twisted discrete series, that is, the discrete components of the space of square integrable sections of a line bundle, given by a unitary character on an abelian extension of $H$.Comment: To appear in GAF

SINGULARITIES OF HOLOMORPHICALLY EXTENDED SPHERICAL FUNCTIONS

1. Motivation and background\ud This is a preliminary account of joint work in progress which serves as an (very) extended abstract for a presentation of one of us ( $E.M$ .O.) in the Awajishima conference on Representation Theory, November 16-19, 2004, Japan.\ud To motivate the questions which we will address we will describe in some detail the beautiful ideas that were initiated by Sarnak [18] and by Bernstein and Reznikov [2], and then further explored by Kr\"otz and Stanton [9].\ud Inspired by Sarnak [18], Bernstein and Reznikov [2] proposed a new method for estimating the coefficients in the expansion of the square of a Maass form on a compact locally symmetric space $Z=\Gamma\backslash X$ (where $X=G/K$ denotes a noncompact Riemannian symmetric space, and\ud $\Gamma\subset G$ is a $co$-compact discrete subgroup of $G$ ) with respect to an orthonormal basis of $L^{2}(Z)$ consisting of Maass forms. The method is based on holomorphic extension of irreducible representations of $G$\ud to a certain $G$-invariant domain in $X_{\mathbb{C}}:=G_{\mathbb{C}}/If_{\mathbb{C}}$ (we assume that $G\subset G_{\mathbb{C}})$ .\ud In [2] the method was applied in the case of $G=SL_{2}(\mathbb{R})$ . The method was carried further by Kr\"otz and Stanton in [9], where the results of [2] were slightly improved for $G=SL_{2}(\mathbb{R})$ , and similar results\ud for other rank 1 Riemannian symmetric spaces $G/K$ were obtained. In addition some higher rank cases were considered in [9]. These considerations gave rise to various interesting issues concerning holomorphic\ud extensions of representations and their matrix coefficients

Atrial Natriuretic Peptide Induces Mitogen-Activated Protein Kinase Phosphatase-1 in Human Endothelial Cells via Rac1 and NAD(P)H Oxidase/Nox2-Activation

The cardiovascular hormone atrial natriuretic peptide (ANP) exerts anti-inflammatory effects on tumor necrosis factor-α–activated endothelial cells by inducing mitogen-activated protein kinase (MAPK) phosphatase-1 (MKP-1). The underlying mechanisms are as yet unknown. We aimed to elucidate the signaling pathways leading to an induction of MKP-1 by ANP in primary human endothelial cells. By using antioxidants, generation of reactive oxygen species (ROS) was shown to be crucially involved in MKP-1 upregulation. ANP was found to increase ROS formation in cultured cells as well as in the endothelium of intact rat lung vessels. We applied NAD(P)H oxidase (Nox) inhibitors (apocynin and gp91ds-tat) and revealed this enzyme complex to be crucial for superoxide generation and MKP-1 expression. Moreover, by performing Nox2/4 antisense experiments, we identified Nox2 as the critically involved Nox homologue. Pull-down assays and confocal microscopy showed that ANP activates the small Rho-GTPase Rac1. Transfection of a dominant-negative (RacN17) and constitutively active Rac1 mutant (RacV12) indicated that ANP-induced superoxide generation and MKP-1 expression are mediated via Rac1 activation. ANP-evoked production of superoxide was found to activate c-Jun N-terminal kinase (JNK). Using specific inhibitors, we linked ANP-induced JNK activation to MKP-1 expression and excluded an involvement of protein kinase C, extracellular signal-regulated kinase, and p38 MAPK. MKP-1 induction was shown to depend on activation of the transcription factor activator protein-1 (AP-1) by using electrophoretic mobility shift assay and AP-1 decoys. In summary, our work provides insights into the mechanisms by which ANP induces MKP-1 and shows that ANP is a novel endogenous activator of endothelial Rac1 and Nox/Nox2

Crucial role of local peroxynitrite formation in neutrophil-induced endothelial cell activation

Introduction and methods: The reaction of superoxide anions and NO not only results in a decreased availability of NO, but also leads to the formation of peroxynitrite, the role of which in the cardiovascular system is still discussed controversially. In cultured human endothelial cells, we studied whether there is a significant interaction between endothelial NO and neutrophil-derived superoxide anions in terms of endothelial peroxynitrite formation. We particularly studied whether a significantly higher redox-stress can be found in those endothelial cells directly adjacent to an activated neutrophil. Results: A considerable part of the 2,7-dihydrodichlorofluoresceine signal in endothelial cells was due to oxidation by peroxynitrite. Providing superoxide radicals by enzymatic source or by the neutrophil respiratory burst increased the fluorescence, which was attenuated by blockade of endothelial NO-synthase, suggesting that peroxynitrite was formed from neutrophil- or extracellular enzyme-derived superoxide and endothelial NO. Considerably higher fluorescence intensity was observed in endothelial cells in direct neighborhood to a neutrophil. This was particularly pronounced in the presence of a NO-donor and was accompanied by a strong activation of NF-κB and increased expression of E-selectin in these cells. Conclusion: Endothelial cells adjacent to neutrophils may have elevated levels of peroxynitrite that result in an increased expression of adhesion molecules. Such cells might represent a preferential site for adhesion and migration of additional neutrophils when simultaneously high concentrations of NO and neutrophil-derived superoxide are present

Ellipticity and discrete series

We explain by elementary means why the existence of a discrete series representation of a real reductive group $G$ implies the existence of a compact Cartan subgroup of $G$. The presented approach has the potential to generalize to real spherical spaces

Inhibition of the tyrosine phosphatase SHP-2 suppresses angiogenesis in vitro and in vivo

Endothelial cell survival is indispensable to maintain endothelial integrity and initiate new vessel formation. We investigated the role of SHP-2 in endothelial cell survival and angiogenesis in vitro as well as in vivo. SHP-2 function in cultured human umbilical vein and human dermal microvascular endothelial cells was inhibited by either silencing the protein expression with antisense-oligodesoxynucleotides or treatment with a pharmacological inhibitor (PtpI IV). SHP-2 inhibition impaired capillary-like structure formation (p < 0.01; n = 8) in vitro as well as new vessel growth ex vivo (p < 0.05; n = 10) and in vivo in the chicken chorioallantoic membrane (p < 0.01, n = 4). Additionally, SHP-2 knock-down abrogated fibroblast growth factor 2 (FGF-2)-dependent endothelial proliferation measured by MTT reduction ( p ! 0.01; n = 12). The inhibitory effect of SHP-2 knock-down on vessel growth was mediated by increased endothelial apoptosis ( annexin V staining, p ! 0.05, n = 9), which was associated with reduced FGF-2-induced phosphorylation of phosphatidylinositol 3-kinase (PI3-K), Akt and extracellular regulated kinase 1/2 (ERK1/2) and involved diminished ERK1/2 phosphorylation after PI3-K inhibition (n=3). These results suggest that SHP-2 regulates endothelial cell survival through PI3-K-Akt and mitogen-activated protein kinase pathways thereby strongly affecting new vessel formation. Thus, SHP-2 exhibits a pivotal role in angiogenesis and may represent an interesting target for therapeutic approaches controlling vessel growth. Copyright (C) 2007 S. Karger AG, Basel

Schreier type theorems for bicrossed products

We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups $(H, G, \alpha, \beta)$ is deformed using a combinatorial datum $(\sigma, v, r)$ consisting of an automorphism $\sigma$ of $H$, a permutation $v$ of the set $G$ and a transition map $r: G\to H$ in order to obtain a new matched pair $\bigl(H, (G,*), \alpha', \beta' \bigl)$ such that there exist an $\sigma$-invariant isomorphism of groups $H {}_{\alpha} \bowtie_{\beta} G \cong H {}_{\alpha'} \bowtie_{\beta'} (G,*)$. Moreover, if we fix the group $H$ and the automorphism \sigma \in \Aut(H) then any $\sigma$-invariant isomorphism $H {}_{\alpha} \bowtie_{\beta} G \cong H {}_{\alpha'} \bowtie_{\beta'} G'$ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications two Schreier type classification theorems for bicrossed product of groups are given.Comment: 21 pages, final version to appear in Central European J. Mat