The Capelli eigenvalue problem

Abstract: In the 1980s Bert Kostant and the speaker studied a family of invariant differential operators $D_\mu$ associated to a Jordan algebra, which generalize the Capelli operator of classical invariant theory.

In the 1990s speaker showed that the eigenvalues of $D_\mu$ are given by a symmetric polynomial $R_\mu$, which can be characterized by certain rather simple vanishing properties. Subsequently, F. Knop and the speaker found a remarkable connection between the $R_\mu$ and Jack polynomials, which enabled us to prove a positivity conjecture of Macdonald.

We will recall these ideas and discuss some more recent developments in the subject. These include results for Lie superalgebras and quantum groups, as well as a new combinatorial formula for the $R_\mu$ which proves an old positivity conjecture due to Knop and the speaker.