The Cone of Hilbert Nullforms

A geometric–combinatorial algorithm is given that allows one, using solely the system of weights and roots, to determine the Hesselink strata of the nullcone of a linear representation of a reductive algebraic group and compute their dimensions. In particular, it provides a constructive approach to computing the dimension of the nullcone and determining all its irreducible components of maximal dimension. In the case of the adjoint representation (and, more generally, $\theta$-representation), the algorithm turns into the algorithm of classifying conjugacy classes of nilpotent elements in a semisimple Lie algebra (respectively, homogeneous nilpotent elements in a cyclically graded semisimple Lie algebra).