Representation Theory of Diffeomorphism Groups - Constructions

Constructions

Let therefore M be a n-dimensional connected differentiable manifold, and x be any point on it. Let Diff(M) be the orientation-preserving diffeomorphism group of M (only the identity component of mappings homotopic to the identity diffeomorphism if you wish) and Diff_x1(M) the stabilizer of x. Then, M is identified as a homogeneous space

Diff(M)/Diff_x1(M).

From the algebraic point of view instead, is the algebra of smooth functions over M and is the ideal of smooth functions vanishing at x. Let be the ideal of smooth functions which vanish up to the n-1th partial derivative at x. is invariant under the group Diff_x1(M) of diffeomorphisms fixing x. For n > 0 the group Diff_xn(M) is defined as the subgroup of Diff_x1(M) which acts as the identity on . So, we have a descending chain

Diff(M) ⊃ Diff_x1(M) ⊃ ... ⊃ Diff_xn(M) ⊃ ...

Here Diff_xn(M) is a normal subgroup of Diff_x1(M), which means we can look at the quotient group

Diff_x1(M)/Diff_xn(M).

Using harmonic analysis, a real- or complex-valued function (with some sufficiently nice topological properties) on the diffeomorphism group can be decomposed into Diff_x1(M) representation-valued functions over M.