of the Capelli identity.

Later on, Howe and Umeda [@HoweUmeda] generalized the Capelli identity to the setting of multiplicity-free spaces by posing and solving two general questions, which they called the *abstract* and *concrete* Capelli problems. Let $\g g$ be a complex reductive Lie algebra, and $W$ be a multiplicity-free $\g g$-space. The abstract Capelli problem asks whether the centre $\bfZ(\g g)$ of $\bfU(\g g)$, the universal enveloping algebra of $\g g$, maps surjectively onto the algebra $\sPD(W)^{\g g}$ of $\g g$-invariant polynomial-coefficient differential operators on $W$. The concrete Capelli problem asks for explicit elements of $\bfZ(\g g)$ whose images generate $\sPD(W)^\g g$.

Around the same time as [@HoweUmeda], Kostant and Sahi [@KostantSahi1], [@KostantSahi2] considered a slightly different question, which we shall refer to here as the Capelli *eigenvalue* problem. It turns out that the algebra $\sPD(W)^{\g g}$ admits a natural basis $D_{\lambda }$ – the *Capelli basis* – which is indexed by the monoid of highest weights of $\g g$-modules $V_{\lambda }$ occurring in the symmetric algebra $\sS\left( W\right) $. The Capelli eignevalue problem asks for determination of the eigenvalue of $D_{\mu }$ on $V_{\lambda }$. It turns out that these eignvalues are of the form $\varphi
_{\mu }\left( 