Geometric Complexity Theory -- Lie Algebraic Methods for Projective Limits of Stable Points

Let G be a connected reductive algebraic group over C, with Lie algebra G, acting rationally on a complex vector space V , and the corresponding projective space PV . Let x ∈ V and let H ⊆ G be its stabilizer and H ⊆ G, its Lie algebra. Our primary objective is to understand the points [y], and their stabilizers, which occur in the vicinity of [x] in PV . Towards this, we construct an explicit Lie algebra action of G on a suitably parametrized neighbourhood of x. As a consequence, we show that the Lie algebras of the stabilizers of points in the neighbourhood of x are parameterized by subspaces of H. When H is reductive, our results imply that these are in fact, Lie subalgebras of H. If the orbit of x were closed this would also follow from a celebrated theorem of Luna [Lun73]. To construct our Lie algebra action we proceed as follows. We identify the tangent space to the orbit Ox of x with a complement ofH in G. Let N be any linear subspace of V that is complementary to TOx, the tangent space to the orbit Ox. We construct an explicit map θ : V × N → V which captures the action of the nilpotent part of H on V . We show that the map θ is intimately connected with the local curvature form of the orbit at that point. We call this data, of a neighbourhood of x with the Lie algebra morphism from G to vector fields in this neighbourhood, a local model at x. The action of G on V extends to an action on P(V ). We illustrate the utility of the local model in understanding when [x] ∈ PV is in the projective orbit closure of [y] ∈ PV via two applications. The first application is when V = Sym(X), the space of forms in the variablesX := {x1, . . . , xk}. We consider a form f (as the “y” above) and elements in its orbit given by f(t) = A(t)·f , where A(t) is an invertible family in GL(X), with t ∈ C. We express f(t) as f(t) = tg + tfb + higher terms, with a < b. We call g ∈ Sym(X), the leading term as the limit point (the ”x” above) and fb as the direction of approach. Let K be the Lie algebra of the stabilizer of f and H the Lie algebra of the stabilizer of g. The local analysis gives us a flattening K0 of K as a subalgebra of H, thereby connecting the two stabilizers. There is a natural action of H on N = V/TOg. We show that K0 also stabilizes fb ∈ N . We show that there is an ǫ-extension Lie algebra K(ǫ) of K0 whose structure Computer Science Department, Indian Institute of Technology, Mumbai, India. adsul@cse.iitb.ac.in Computer Science Department, Indian Institute of Technology, Mumbai, and Indian Institute of Technology, Goa, India. sohoni@cse.iitb.ac.in Chennai Mathematical Institute, Chennai, India. kv@cmi.ac.in