A quite simple method for determining the mod 2 cohomology of the compact, connected, simple, exceptional Lie group of rank 8 is given.

Let $\Omega = G/K$ be a bounded symmetric domain in a complex vector space $V$ with the Lebesgue measure $dm(z)$ and the Bergman reproducing kernel $h(z,w)^{-p}$. Let $d\mu _{\alpha}(z) = h(z, \bar{z})^{\alpha}dm(z)$, $\alpha > -1$, be the weighted measure on $\Omega$. The group $G$ acts unitarily on the space $L^{2}(\Omega , \mu_\alpha )$ via change of variables together with a multiplier. We consider the discrete parts, also called the relative discrete series, in the irreducible decomposition of the $L^{2}$-space. Let $\bar{D} = B(z, \bar{z})\partial$ be the invariant Cauchy-Riemann operator. We realize the relative discrete series as the kernels of the power $\bar{D}^{m+1}$ of the invariant Cauchy-Riemann operator $\bar{D}$ and thus as nearly holomorphic functions in the sense of Shimura. We prove that, roughly speaking, the operators $\bar{D}^{m}$ are intertwining operators from the relative discrete series into the standard modules of holomorphic discrete series (as Bergman spaces of vector-valued holomorphic functions on $\Omega$).

This is a continuation of the previous work [CKMS01] by the authors on classification of equivariant complex vector bundles over a circle. In this paper equivariant real vector bundles over a circle with a compact Lie group action are classified, by characterizing their fiber representations, and by using the result of the complex case. Their triviality is also treated. The basic phenomenon is similar to the complex case but more complicated here.

We classify binary self-similar sets, which are compact sets determined by two contractions on the plane, into four classes from the viewpoint of functional equations. In this classification, we can not only show close relationships among functions with self-similarity but also give solutions to a few open problems in other field.

In this paper we prove that over local or global fields of characteristic 0, the Corestriction Principle holds for kernel and image of all maps which are connecting maps in group cohomology which extends an earlier result due to Deligne and can be considered as cohomological counterpart to a result of Lenstra and Tate.

In this paper we prove that certain natural birational and arithmetic invariants of connected subgroups of linear algebraic groups all defined over a local or global field of characteristic 0 are bounded in terms of the ambient group and the base field.

Let $\Omega$ be a smoothly bounded pseudoconvex domain in $\mathbb{C}^{n}$ and let $z_{0} \in b\Omega$ be a point of finite type. We also assume that the Levi form of $b\Omega$ is comparable in a neighborhood of $z_{0}$. Then we get a quantity which bounds from above and below the Bergman metric, Caratheodory metric and Kobayashi metric in a small constant and large constant sense.

Renormalization plays a very important role in the study of the dynamics of quadratic polynomials. We generalize renormalization to polynomials of any degree so that a lot of known properties still hold for the generalized renormalization. In particular, we generalize McMullen’s result that any robust infinitely renormalizable quadratic polynomial carries no invariant line field.

We construct finite composition series of group $C^{*}$-algebras of the generalized Mautner groups whose subquotients are tensor products of commutative $C^{*}$-algebras, noncommutative tori and the $C^{*}$-algebra of compact operators. As an application, we estimate the stable rank and connected stable rank of the $C^{*}$-algebras of generalized real Mautner groups.