Representation Theory of Diffeomorphism Groups - The Supply of Representations

The Supply of Representations

So what are the reps of Diff_x1(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

Diff_x1(M)/Diff_xn(M),

we can obtain a rep of Diff_x1(M).

Let's look at

Diff_x1(M)/Diff_x2(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)?

.

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

Diff_x1(M)/Diff_xn(M).

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

Diff_x1(M)/Diff_x1(M) ⊂ ... ⊂ Diff_x1(M)/Diff_xn(M) ⊂ ...

Here, the "⊂" 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

Diff_x1(M)/Diff_xm(M)

can automatically be turned into a rep of

Diff_x1/Diff_xn(M)

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

Diff_x1/Diff_xp + 2

which doesn't arise from a rep of

Diff_x1/Diff_xp + 1.

Then, we call the fiber bundle with that rep as the fiber (i.e. Diff_x1/Diff_xp + 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 Diff_x1(M) and the larger group being Diff(M).