Representation theory of diffeomorphism groups

In mathematics, a source for the representation theory of the group of diffeomorphisms of a smooth manifold M is the initial observation that (for M connected) that group acts transitively on M.

History
A survey paper from 1975 of the subject by Anatoly Vershik, Israel Gelfand and M. I. Graev attributes the original interest in the topic to research in theoretical physics of the local current algebra, in the preceding years. Research on the finite configuration representations was in papers of R. S. Ismagilov (1971), and A. A. Kirillov (1974). The representations of interest in physics are described as a cross product C∞(M)·Diff(M).

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 Diffx1(M) the stabilizer of x. Then, M is identified as a homogeneous space


 * Diff(M)/Diffx1(M).

From the algebraic point of view instead, $$C^\infty(M)$$ is the algebra of smooth functions over M and $$I_x(M)$$ is the ideal of smooth functions vanishing at x. Let $$I_x^n(M)$$ be the ideal of smooth functions which vanish up to the n-1th partial derivative at x. $$I_x^n(M)$$ is invariant under the group Diffx1(M) of diffeomorphisms fixing x. For n > 0 the group Diffxn(M) is defined as the subgroup of Diffx1(M) which acts as the identity on $$I_x(M)/I_x^n(M)$$. So, we have a descending chain


 * Diff(M) &sup; Diffx1(M) &sup; ... &sup; Diffxn(M) &sup; ...

Here Diffxn(M) is a normal subgroup of Diffx1(M), which means we can look at the quotient group


 * Diffx1(M)/Diffxn(M).

Using harmonic analysis, a real- or complex-valued function (with some sufficiently nice topological properties) on the diffeomorphism group can be decomposed into Diffx1(M) representation-valued functions over M.

The supply of representations
So what are the representations of Diffx1(M)? Let's use the fact that if we have a group homomorphism φ:G → H, then if we have a H-representation, we can obtain a restricted G-representation. So, if we have a rep of


 * Diffx1(M)/Diffxn(M),

we can obtain a rep of Diffx1(M).

Let's look at


 * Diffx1(M)/Diffx2(M)

first. This is isomorphic to the general linear group GL+(n, R) (and because we're only considering orientation preserving diffeomorphisms and so the determinant is positive). What are the reps of GL+(n, R)?


 * $$GL^+(n,\mathbb{R})\cong \mathbb{R}^+\times SL(n,\mathbb{R})$$.

We know the reps of SL(n, R) are simply tensors over n dimensions. How about the R+ part? That corresponds to the density, or in other words, how the tensor rescales under the determinant of the Jacobian of the diffeomorphism at x. (Think of it as the conformal weight if you will, except that there is no conformal structure here). (Incidentally, there is nothing preventing us from having a complex density).

So, we have just discovered the tensor reps (with density) of the diffeomorphism group.

Let's look at


 * Diffx1(M)/Diffxn(M).

This is a finite-dimensional group. We have the chain


 * Diffx1(M)/Diffx1(M) &sub; ... &sub; Diffx1(M)/Diffxn(M) &sub; ...

Here, the "&sub;" signs should really be read to mean an injective homomorphism, but since it is canonical, we can pretend these quotient groups are embedded one within the other.

Any rep of


 * Diffx1(M)/Diffxm(M)

can automatically be turned into a rep of


 * Diffx1/Diffxn(M)

if n > m. Let's say we have a rep of


 * Diffx1/Diffxp + 2

which doesn't arise from a rep of


 * Diffx1/Diffxp + 1.

Then, we call the fiber bundle with that rep as the fiber (i.e. Diffx1/Diffxp + 2 is the structure group) a jet bundle of order p.

Side remark: This is really the method of induced representations with the smaller group being Diffx1(M) and the larger group being Diff(M).

Intertwining structure
In general, the space of sections of the tensor and jet bundles would be an irreducible representation and we often look at a subrepresentation of them. We can study the structure of these reps through the study of the intertwiners between them.

If the fiber is not an irreducible representation of Diffx1(M), then we can have a nonzero intertwiner mapping each fiber pointwise into a smaller quotient representation. Also, the exterior derivative is an intertwiner from the space of differential forms to another of higher order. (Other derivatives are not, because connections aren't invariant under diffeomorphisms, though they are covariant.) The partial derivative isn't diffeomorphism invariant. There is a derivative intertwiner taking sections of a jet bundle of order p into sections of a jet bundle of order p + 1.