Let's fix the ground field $\mathbb{C}$.
In the paper "Shifted Schur Functions" http://arxiv.org/abs/q-alg/9605042 Andrei Okounkov and Grigori Olshanski introduce a special basis for the center $Z(\mathfrak{gl}_n)$ of the enveloping algebra $U(\mathfrak{gl}_n)$ which has some nice algebraic properties which they call "coherence properties". This enables them to prove Capelli type identies.
I am wondering if there is an equivalent basis for $\mathfrak{sl}_n$.
