Polynomiality of invariants, unimodularity and adapted pairs

Let mathfraka mathfrak{a} be a finite-dimensional Lie algebra and Y( mathfraka ) Yleft( mathfrak{a} right) the mathfraka mathfrak{a} invariant subalgebra of its symmetric algebra S( mathfraka ) Sleft( mathfrak{a} right) under adjoint action. Recently there has been considerable interest in studying situations when Y( mathfraka ) Yleft( mathfrak{a} right) may be polynomial on index mathfraka mathfrak{a} generators, for example if mathfraka mathfrak{a} is a biparabolic or a centralizer mathfrakg^x {mathfrak{g}^x} in a semisimple Lie algebra mathfrakg mathfrak{g} .