Geometric Complexity Theory – Lie Algebraic Methods for Projective Limits of Stable Points
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Let G be a connected reductive group acting on a complex vector space V and projective space ℙV. Let x∈ V and H⊆ G be the Lie algebra of its stabilizer. Our objective is to understand points [y], and their stabilizers which occur in the vicinity of [x]. We construct an explicit G-action on a suitable neighbourhood of x, which we call the local model at x. We show that Lie algebras of stabilizers of points in the vicinity of x are parameterized by subspaces of H. When H is reductive these are Lie subalgebras of H. If the orbit of x is closed this also follows from Luna's theorem. Our construction involves a map connected to the local curvature form at x. We apply the local model to forms, when the form g is obtained from the form f as the leading term of a one parameter family acting on f. We show that there is a flattening K_0 of K, the stabilizer of f which sits as a subalgebra of H, the stabilizer g. We specialize to the case of forms f whose SL(X)-orbits are affine, and the orbit of g is of co-dimension 1. We show that (i) either H has a very simple structure, or (ii) conjugates of the elements of K also stabilize g and the tangent of exit. Next, we apply this to the adjoint action. We show that for a general matrix X, the signatures of nilpotent matrices in its projective orbit closure (under conjugation) are determined by the multiplicity data of the spectrum of X. Finally, we formulate the path problem of finding paths with specific properties from y to its limit points x as an optimization problem using local differential geometry. Our study is motivated by Geometric Complexity Theory proposed by the second author and Ketan Mulmuley.
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