GCT and symmetries

Let A be a finite dimensional complex vector space with a polynomial action of G = GLk. One main example is A = Sym C. (For tensors G = GLk, A = ⊗C) Let Z ⊆ A be locally closed. One main example is a group orbit Z = Gv. Recall (last semester, Chapter 15) that for a locally closed set Z ⊆ A we defined the coordinate ring C[Z] as the ring of regular functions (i.e., locally defined by rational functions) on Z. Therefore C[Z] ⊆ C[Z], because C[Z] is the ring of polynomial functions on Z, and polynomial functions are regular. For a point v ∈ A let stabG(v) ⊆ G denote its stabilizer (or its symmetry group): stabG(v) := {g ∈ G | gv = v}.