Let $G$ be a simply connected nilpotent Lie group, $\mathcal{G}$ the finite-dimensional Lie algebra of $G$, $\mathcal{V}$ a finite-dimensional vector space over $\mathbb{C}$ or $\mathbb{R}$, and $H$ a connected Lie subgroup of $G$ such that the Lie algebra of $H$ is a subordinate subalgebra to an element $\pi $ of $Hom\left( \mathcal{G},gl\left( \mathcal{V}\right) \right) $. In this work, we construct an irreducible representation $\chi _{\pi }$ of $H$ such that the induced of $ \chi _{\pi }$ on $G$ is irreducible.

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. We show that a complete noncompact manifold $M$ with asymptotically nonnegative Ricci curvature and sectional curvature $K(x)\ge \frac{C}{d_{p}(x)^{\alpha}}$ is diffeomorphic to the Euclidean space $\mathbb{R}^n$ under some conditions on the density of rays starting from the base point $p$ or on the volume growth of geodesic balls in $M$.

In this paper, we present some inequalities for the growth and derivatives of a polynomial with zeros outside a circle of arbitrary radius $k\gt 0$. Our results provide improvements and generalizations of some well known polynomial inequalities.