Differential Operator Specializations of Noncommutative Symmetric Functions

Let $K$ be any unital commutative $\mathbb{Q}$-algebra and $z=(z_1,\mathrm{.}\mathrm{.}\mathrm{.},z_n)$ commutative or noncommutative free variables. Let $t$ be a formal parameter which commutes with $z$ and elements of $K$. We denote uniformly by $\text{kzz}$ and $\text{kttzz}$ the formal power series algebras of $z$ over $K$ and $K[[t]]$, respectively. For any $\alpha \ge 1$, let $\text{cDazz}$ be the unital algebra generated by the differential operators of $\text{kzz}$ which increase the degree in $z$ by at least $\alpha -1$ and $\text{ataz}$ the group of automorphisms $F_t(z)=z-H_t(z)$ of $\text{kttzz}$ with $o(H_t(z))\ge \alpha$ and $H_{t=0}(z)=0$. First, for any fixed $\alpha \ge 1$ and $F_t\in \text{ataz}$, we introduce five sequences of differential operators of $\text{kzz}$ and show that their generating functions form a $\mathcal{N}$CS (noncommutative symmetric) system [Z4] over the differential algebra $\text{cDazz}$. Consequently, by the universal property of the $\mathcal{N}$CS system formed by the generating functions of certain NCSFs (noncommutative symmetric functions) first introduced in [GKLLRT], we obtain a family of Hopf algebra homomorphisms ${\text{cS}}_{F_t}:\mathcal{N}Sym\to \text{cDazz}$ $(F_t\in \text{ataz})$, which are also grading-preserving when $F_t$ satisfies certain conditions. Note that, the homomorphisms ${\text{cS}}_{F_t}$ above can also be viewed as specializations of NCSFs by the differential operators of $\text{kzz}$. Secondly, we show that, in both commutative and noncommutative cases, this family ${\text{cS}}_{F_t}$ (with all $n\ge 1$ and $F_t\in \text{ataz}$) of differential operator specializations can distinguish any two different NCSFs. Some connections of the results above with the quasi-symmetric functions ([Ge], [MR], [S]) are also discussed.