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Detecting orbits along subvarieties via the moment map
Let G be a (real or complex) linear reductive algebraic group acting on an affine variety V . Let W be a subvariety. In this work we study how the G-orbits intersect W. We develop a criterion to determine when the intersection can be described as a finite union of orbits of a reductive subgroup. The conditions of the criterion are easily verified in practice and are used to develop techniques to study left-invariant Ricci soliton metrics on nilpotent Lie groups. A nilpotent Lie group is called an Einstein nilradical if it admits a left-invariant Ricci soliton metric. Applying the techniques developed, we show that the classification of Einstein nilradicals can be reduced to the class of so-called indecomposable groups. Among other applications, we construct arbitrarily large continuous families of (non-isomorphic) nilpotent Lie groups which do not admit left-invariant Ricci soliton metrics. The note finishes by applying our techniques to the adjoint representation of reductive Lie groups. The classical result of finiteness of nilpotent orbits is reproved and it is shown that each of these orbits contains a critical point of the norm squared of the moment map.