A generalization of Wilkerson’s mod $p$ decompositions of compact, connected, semi-simple Lie groups [7] to the non-simply connected case is given. As an application, H-spaces with interesting fundamental groups are constructed.

We determine the structure of category $\mathcal{O}$ for the rational Cherednik algebra of the wreath product complex reflection group $G(m,1,n)$ in the case where the $\mathsf{KZ}$ functor satisfies a condition called separating simples. As a consequence, we show that the property of having exactly $N-1$ simple modules, where $N$ is the number of simple modules of $G(m,1,n)$, determines the Ariki-Koike algebra up to isomorphism.

Let $\mathfrak{g}$ be a complex simple Lie algebra. We have the adjoint representation of the adjoint group $G$ on $\mathfrak{g}$. Then $G$ acts on the projective space $\mathbb{P}_{\mathfrak{g}}$. We consider the closure $X$ of the image of a nilpotent orbit in $\mathbb{P}_{\mathfrak{g}}$. The $i$-secant variety $Sec^{(i)}X$ of a projective variety $X$ is the closure of the union of projective subspaces of dimension $i$ in the ambient space $\mathbb{P}$ spanned by $i+1$ points on $X$. In particular we call the 1-secant variety the secant variety. In this paper we give explicit descriptions of the secant and the higher secant varieties of nilpotent orbits of complex classical simple Lie algebras.

We define and compute the energy of 1-foliations on riemannian manifolds. We then derive the Euler-Lagrange equations associated with the energy. We also prove that Riemannian flows on manifolds of constant curvature are ritical if and only if they are isometric. Finally we prove that isometric flows on 3-manifolds are critical if and only if either they are transverse to 2-dimensional foliations or they provide K-contact structures.

Let $\pi : Y \longrightarrow X$ be a domain over a complex space $X$. Assume that $\pi$ is locally Stein. Then we show that $Y$ is Stein provided that $X$ is Stein and either there is an open set $W$ containing $X_{\mathrm{sing}}$ with $\pi^{-1}(W)$ Stein or $\pi$ is locally hyperconvex over any point in $X_{\mathrm{sing}}$. In the same vein we show that, if $X$ is $q$-complete and $X$ has isolated singularities, then $Y$ results $q$-complete.

We study the representation theory of the rational Cherednik algebra $H_{\kappa}=H_{\kappa}(\mathbb{Z}_{l})$ for the cyclic group $\mathbb{Z}_{l}= \mathbb{Z}/l\mathbb{Z}$ and its connection with the geometry of the quiver variety $\mathfrak{M}_{\theta}(\delta )$ of type $A^{(1)}_{l-1}$.

We consider a functor between the categories of $H_{\kappa}$-modules with different parameters, called the shift functor, and give the condition when it is an equivalence of categories.

We also consider a functor from the category of $H_{\kappa}$-modules with good filtration to the category of coherent sheaves on $\mathfrak{M}_{\theta}(\delta )$. We prove that the image of the regular representation of $H_{\kappa}$ by this functor is the tautological bundle on $\mathfrak{M}_{\theta}(\delta )$. As a corollary, we determine the characteristic cycles of the standard modules. It gives an affirmative answer to a conjecture given in [Go] in the case of $\mathbb{Z}_{l}$.

The Smith equivalence of real representations of a finite group has been studied by many mathematicians, e.g. J. Milnor, T. Petrie, S. Cappell-J. Shaneson, K. Pawałowski-R. Solomon. For a given finite group, let the primary Smith set of the group be the subset of real representation ring consisting of all differences of pairs of prime matched, Smith equivalent representations. The primary Smith set was rarely determined for a nonperfect group G besides the case where the primary Smith set is trivial. In this paper we determine the primary Smith set of an arbitrary Oliver group such that a Sylow 2-subgroup is normal and the nilquotient is isomorphic to the direct product of a finite number of cyclic groups of order 2 or 3. In particular, we answer to a problem posed by T. Sumi.

In the article, established are necessary and sufficient conditions such that the extended mean values are Schur-convex and Schur-concave.