  An enhanced algebraic group $\uG$ of $G=\GL(V)$ over $\bbc$ is a product
variety $\GL(V)\times V$, endowed with an enhanced cross product. Associated
with a natural tensor representation of $\uG$, there are naturally Levi and
parabolic Schur algebras $\mathcal{L}$ and $\mathcal{P}$ respectively. We
precisely investigate their structures, and study the dualities on the enhanced
tensor representations for variant groups and algebras. In this course, an
algebraic model of so-called degenerate double Hecke algebras (DDHA) is
produced, and becomes a powerful implement. The connection between
$\mathcal{L}$ and DDHA gives rise to two results for the classical
representations of $\GL(V)$: (i) A duality between $\GL(V)\times\Gm$ and DDHA
where $\Gm$ is the one-dimensional multiplicative group; (ii) A branching
duality formula. With aid of the above discussion, we further obtain a
parabolic Schur-Weyl duality for $\uG\rtimes \Gm$. What is more, the parabolic
Schur subalgebra turns out to have only one block. The Cartan invariants for
this algebra are precisely determined.
