We prove some new results on Corestriction principle for non-abelian cohomology of group schemes over local and global fields or the rings of integers thereof.

Let $p$ be an odd prime which splits completely into distinct primes in a CM field $K$. By considering ray class field of $K$ with respect to prime ideals lying above $p$, one can define a certain special non-cyclotomic $\mathbf{Z}_{p}$-extension over $K$. We will give some examples of such non-cyclotomic $\mathbf{Z}_{p}$-extensions whose Iwasawa $λ$- and $µ$-invariants both vanish, using a variant of a criterion due to Greenberg.

Let $\varrho : G \to GL(V)$ be a finite dimensional rational representation of a diagonalizable algebraic group $G$ over an algebraically closed field $K$ of characteristic zero. Using a minimal paralleled linear hull $(W, w)$ of $\varrho$ defined in [N4], we show the existence of a cofree representation $\widetilde{G_w} \hookrightarrow GL(W)$ such that $\varrho(G_w) \subseteq \widetilde{G_w}$ and $W//G_w \to W// \widetilde{G_w}$ is divisorially unramified is equivalent to the Gorensteinness of $V//G$.

In the previous paper [N3], we introduced a cofree embedding of a finite dimensional rational representation of a diagonalizable algebraic group over an algebraically closed field of characteristic zero. Using this, we show inductive characterization of toric locally complete intersection singularities in terms of group representation theory. Consequently we obtain a classification of affine monomial normal hypersurfaces.

We develop a group-theoretic method of generalizing the Laplace-Beltrami operators on the classical domains. In [18], we defined the generalized Poisson-Cauchy transforms on the classical domains. We show that the generalized Poisson-Cauchy transforms give us eigenfunctions of the generalized Laplacians defined in this paper.

A Lagrangian submanifold in the complex Euclidean $n$-space ${\bf C}^n$ is called Hamiltonian-stationary if it is a critical point of the area functional restricted to (compactly supported) Hamiltonian variations. In this article, we classify the family of Hamiltonian-stationary Lagrangian submanifolds of ${\bf C}^n$ which are Lagrangian $H$-umbilical.