Geodesic
The Cone of Hilbert Nullforms
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and algebraic geometry, Tome 241 (2003), pp. 192-209

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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). 
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     author = {V. L. Popov},
     title = {The {Cone} of {Hilbert} {Nullforms}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     publisher = {mathdoc},
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     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2003_241_a10/}
}
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V. L. Popov. The Cone of Hilbert Nullforms. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and algebraic geometry, Tome 241 (2003), pp. 192-209. http://geodesic.mathdoc.fr/item/TM_2003_241_a10/