  We study a one-parameter family of vector-valued polynomials associated to
each simple Lie algebra. When this parameter $q$ equals -1 one recovers Joseph
polynomials, whereas at $q$ cubic root of unity one obtains ground state
eigenvectors of some integrable models with boundary conditions depending on
the Lie algebra; in particular, we find that the sum of its entries is related
to numbers of Alternating Sign Matrices and/or Plane Partitions in various
symmetry classes.
