Generalized adjoint actions

Abstract

The aim of this paper is to generalize the classical formula $e^{x}y{e}^{-x}=\sum _{k\ge 0}\frac{1}{k!}(adx{)}^k(y)$ by replacing $e^x$ with any formal power series $f(x)=1+\sum _{k\ge 1}{a}_k{x}^k$. We also obtain combinatorial applications to $q$-exponentials, $q$-binomials, and Hall-Littlewood polynomials.

5 pages, LaTeX, typos corrected, to appear in Journal of Lie Theory