Theory of prehomogeneous vector spaces without regularity condition (Q1187172)

This comprehensive article is based on a series of lectures by the author and deals with complex prehomogeneous vector spaces with respect to a connected reductive group. The author studies relative invariant polynomial functions \(f\) without the standard regularity condition \(\text{det}\left({\partial^ 2\log f\over\partial x_ i\partial x_ j}\right)\not\equiv 0\). The main result says that, like in the regular case, the Fourier transform of a complex power \(f^ \alpha\) is a product of the form \(h\cdot f^{-\alpha}\). But unlike the situation in the regular case, \(h\) is in general supported by the closure of a lower dimensional orbit. The methods and results are strongly related to the theory of \(D\)-modules.