Differential Operator Specializations of Noncommutative Symmetric Functions

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