On enhanced reductive groups (I): Parabolic Schur algebras and the dualities related to degenerate double Hecke algebras

Abstract

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.

35 pages. Some wrong arguments and statemens are corrected. The title is changed. Any comments are welcome