The real spectrum compactification of character varieties: characterizations and applications

We announce results on a compactification of general character varieties that has good topological properties and give various interpretations of its ideal points. We relate this to the Weyl chamber length compactification and apply our results to the theory of maximal and Hitchin representations

Espaces de repr\'esentations compl\`etement r\'eductibles

We study some geometric properties of actions on nonpositively curved spaces related to complete reducibility and semisimplicity, focusing on representations of a finitely generated group in the group G of rational points of a reductive group over a local field, acting on the associated space (symmetric space or affine building). We prove that the space of completely reducible classes is the maximal Hausdorff quotient space for the conjugacy action of G on the representation space.Comment: 15 page

Use of high-plex data provides novel insights into the temporal artery processes of giant cell arteritis

ObjectiveTo identify the key coding genes underlying the biomarkers and pathways associated with giant cell arteritis (GCA), we performed an in situ spatial profiling of molecules involved in the temporal arteries of GCA patients and controls. Furthermore, we performed pharmacogenomic network analysis to identify potential treatment targets.MethodsUsing human formalin-fixed paraffin-embedded temporal artery biopsy samples (GCA, n = 9; controls, n = 7), we performed a whole transcriptome analysis using the NanoString GeoMx Digital Spatial Profiler. In total, 59 regions of interest were selected in the intima, media, adventitia, and perivascular adipose tissue (PVAT). Differentially expressed genes (DEGs) (fold-change > 2 or < −2, p-adjusted < 0.01) were compared across each layer to build a spatial and pharmacogenomic network and to explore the pathophysiological mechanisms of GCA.ResultsMost of the transcriptome (12,076 genes) was upregulated in GCA arteries, compared to control arteries. Among the screened genes, 282, 227, 40, and 5 DEGs were identified in the intima, media, adventitia, and PVAT, respectively. Genes involved in the immune process and vascular remodeling were upregulated within GCA temporal arteries but differed across the arterial layers. The immune-related functions and vascular remodeling were limited to the intima and media.ConclusionThis study is the first to perform an in situ spatial profiling characterization of the molecules involved in GCA. The pharmacogenomic network analysis identified potential target genes for approved and novel immunotherapies

INVARIANT WEAKLY CONVEX COCOMPACT SUBSPACES FOR SURFACE GROUPS IN A 2-BUILDINGS

This paper deals with non-Archimedean representations of punctured surface groups in PGL3, associated actions on (not necessarily discrete) Euclidean buildings of type A2, and degenerations of convex real projective structures on surfaces. The main result is that, under good conditions on Fock-Goncharov generalized shear parameters , non-Archimedean representations acting on the Euclidean building preserve a cocompact weakly convex subspace, which is part flat surface and part tree. In particular the eigenvalue and length(s) spectra are given by an explicit finite A2-complex. We use this result to describe degenerations of convex real projective structures on surfaces for an open cone of parameters. The main tool is a geometric interpretation of Fock-Goncharov parameters in A2-buildings

On triples of ideal chambers in A2-buildings

International audienceWe investigate the geometry in a real Euclidean building X of type A2 of some simple configurations in the associated projective plane at infinity P, seen as ideal configurations in X, and relate it with the projective invariants (from the cross ratio on P). In particular we establish a geometric classification of generic triples of ideal chambers of X and relate it with the triple ratio of triples of flags

Dégénérescences de sous-groupes discrets de groupes de Lie semisimples et actions de groupes sur des immeubles affines

ON ETUDIE ICI LES DEGENERESCENCES DE REPRESENTATIONS FIDELES ET DISCRETES D'UN GROUPE DE TYPE FINI DANS UN GROUPE DE LIE SEMISIMPLE REEL G. ON DONNE D'ABORD DES PROPRIETES FONDAMENTALES DES IMMEUBLES DE TITS AFFINES, LES RELATIONS ENTRE LEURS DIFFERENTES DEFINITIONS APPARAISSANT DANS LA LITTERATURE, ET DE NOUVELLES CARACTERISATIONS PRATIQUES. ON DEMONTRE UNE CLASSIFICATION DE LEURS ISOMETRIES : ELLES FIXENT UN POINT OU TRANSLATENT UNE GEODESIQUE DANS LE COMPLETE. ON DONNE LA CONSTRUCTION PAR LES NORMES ULTRAMETRIQUES DE L'IMMEUBLE DE BRUHAT-TITS DU GROUPE LINEAIRE GL N(F) SUR UN CORPS VALUE F QUELCONQUE. ON DEMONTRE QU'UN SOUS-GROUPE DE TYPE FINI DE GL N(F), DONT TOUT ELEMENT FIXE UN POINT DANS , ADMET UN POINT FIXE GLOBAL DANS LE COMPLETE DE . ON CONSIDERE ENSUITE L'ESPACE X(, G) DES CLASSES DE CONJUGAISON DE REPRESENTATIONS FIDELES ET DISCRETES DE DANS G, TELLES QUE LES ACTIONS CORRESPONDANTES DE SUR L'ESPACE SYMETRIQUE X = G/K N'ADMETTENT PAS DE POINT FIXE GLOBAL A L'INFINI. ON DEFINIT LE VECTEUR DE TRANSLATION D'UN ELEMENT G DE G COMME L'UNIQUE VECTEUR DE LONGUEUR MINIMALE ADHERENT A L'ENSEMBLE DES PROJECTIONS DANS UNE CHAMBRE DE WEYL FERMEE FIXEE $ DE X DES SEGMENTS JOIGNANT UN POINT DE X A SON IMAGE PAR G. ON CONSTRUIT, PAR DES METHODES PUREMENT GEOMETRIQUES, UNE COMPACTIFICATION DE X(, G), INDUITE PAR LE SPECTRE MARQUE DES VECTEURS DE TRANSLATION, GENERALISANT CELLE DE THURSTON POUR L'ESPACE DE TEICHMULLER. ON MONTRE QUE LES POINTS DU BORD SONT LES SPECTRES MARQUES DE VECTEURS DE TRANSLATION SOIT DE REPRESENTATIONS FIDELES ET DISCRETES DE DANS G AYANT UN POINT FIXE GLOBAL A L'INFINI DANS X, SOIT D'ACTIONS DE SUR UN IMMEUBLE AFFINE, QUE L'ON EXPLICITE. LORSQUE EST UN GROUPE DE SURFACE ET G = SL 3(R), CECI DONNE UNE COMPACTIFICATION DE LA COMPOSANTE DE HITCHIN DE L'ESPACE DES MODULES DE G-FIBRES PLATS SUR LA SURFACE, DONT ON CALCULE EXPLICITEMENT LE SPECTRE MARQUE DES VECTEURS DE TRANSLATION DE CERTAINS POINTS DU BORD.ORSAY-PARIS 11-BU Sciences (914712101) / SudocORSAY-PARIS 11-Bib. Maths (914712203) / SudocSudocFranceF