Based on the hereditary property from “mother groups” $G(m,1,n)$, $4\le n\le \mathrm{\infty }$, to their “child groups” $G(m,p,n)$, $p|m,p>1$, the so-called complex reflection groups studied in previous work with E. Hirai, we provide a detailed study of the classification and construction first of irreducible projective representations (i.e., spin representations) and their characters of the generalized symmetric groups $G(m,1,n)$, and then of spin characters of the inductive limit groups $G(m,1,\mathrm{\infty })$. By further studying the heredity, we give the main kernel of the results for the child groups with $p|m,p>1$.

We show a Cartan decomposition for a reductive real spherical homogeneous space.

We consider the assembly map for principal bundles with fiber a countable discrete group. We obtain an index-theoretic interpretation of this homomorphism by providing a tensor-product presentation for the module of sections associated to the Miščenko line bundle. In addition, we give a proof of Atiyah’s $L^2$-index theorem in the general context of flat bundles of finitely generated projective Hilbert $C^{\ast }$-modules over compact Hausdorff spaces. We thereby also reestablish that the surjectivity of the Baum–Connes assembly map implies the Kadison–Kaplansky idempotent conjecture in the torsion-free case. Our approach does not rely on geometric K-homology but rather on an explicit construction of Alexander–Spanier cohomology classes coming from a Chern character for tracial function algebras.

Our aim in this paper is to discuss the weak estimate for the maximal and potential operators in the nonhomogeneous central Morrey-type space $M^{p(\cdot ),q,\mathrm{\nu }}({\mathbf{R}}^n)$, which consists of all measurable functions f on ${\mathbf{R}}^n$ satisfying

$$\Vert f{\Vert }_{M^{p(\cdot ),q,\mathrm{\nu }}({\mathbf{R}}^n)}={\left({\int }_1^{\mathrm{\infty }}{(r^{-\mathrm{\nu }}\Vert f{\Vert }_{L^{p(\cdot )}(B(0,r))})}^q\frac{dr}{r}\right)}^{1∕q}<\mathrm{\infty },$$

where $\mathrm{\nu }>0$, $0<q<\mathrm{\infty }$ and $p(\cdot )$ is a variable exponent satisfying the log-Hölder condition at infinity. When $q=\mathrm{\infty }$, we apply a necessary modification.

Associated to a symplectic quotient $M∕∕G$ is a Lagrangian correspondence ${\mathrm{\Lambda }}_G$ from $M∕∕G$ to M. In this note, we construct in two examples quilts with seam condition on such a correspondence, in the case of $S^1$ acting on ${\mathbb{CP}}^2$ with symplectic quotient ${\mathbb{CP}}^2∕∕S^1={\mathbb{CP}}^1$. First, we exhibit the moduli space of quilted strips that would, if not for figure-eight bubbling, identify the Floer chain groups $\mathit{CF}(\mathit{\gamma },S_{\mathrm{Cl}}^1)$ and $\mathit{CF}({\mathbb{RP}}^2,T_{\mathrm{Cl}}^2)$, where γ is the connected double-cover of ${\mathbb{RP}}^1$. Second, we answer a question due to Akveld–Cannas da Silva–Wehrheim by explicitly producing a figure eight bubble which obstructs an isomorphism between two Floer chain groups. The figure-eight bubbles that we construct in this paper are the first concrete examples of this phenomenon.

We completely classify the $\mathbb{Q}$-factorial terminal toric Fano 3-folds such that the sum of the squared torus invariant prime divisors is non-negative.

In this paper, we study the absolute anabelian geometry of hyperbolic orbicurves. The first main result of the present paper shows the absolute version of the Grothendieck conjecture for quasi-tripods (e.g., hyperbolic curves of genus less than two) over, for instance, finitely generated extensions of mixed-characteristic local fields. Moreover, we prove some absolute anabelian results for certain hyperbolic polycurves as applications of the first main result. Finally, we also show the absolute version of the Grothendieck conjecture for MLF-isotrivial hyperbolic orbicurves over finitely generated extensions of mixed-characteristic local fields.

We give a short and elementary proof of the fact that every metric space of finite asymptotic dimension can be coarsely embedded into a finite product of trees.