# Representation theory of diffeomorphism groups

In

mathematics , a source for therepresentation theory of the group ofdiffeomorphism s of asmooth 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 andM. I. Graev attributes the original interest in the topic to research intheoretical physics of thelocal current algebra , in the preceding years. Research on the "finite configuration" representations was in papers ofR. S. Ismagilov (1971), andA. 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-preservingdiffeomorphism group of "M" (only theidentity component of mappingshomotopic to the identity diffeomorphism if you wish) and Diff_{"x"}^{1}("M") the stabilizer of "x". Then, "M" is identified as ahomogeneous space :Diff("M")/Diff

_{"x"}^{1}("M").From the algebraic point of view instead, $C^infty(M)$ is the algebra of

smooth function s over "M" and $I\_x(M)$ is theideal of smooth functions vanishing at "x". Let $I\_x^n(M)$ be the ideal of smooth functions which vanish up to the n-1^{th}partial derivative at "x". $I\_x^n(M)$ is invariant under the group Diff_{"x"}^{1}("M") of diffeomorphisms fixing x. For "n" > 0 the group Diff_{"x"}^{"n"}("M") is defined as thesubgroup of Diff_{"x"}^{1}("M") which acts as the identity on $I\_x(M)/I\_x^n(M)$. So, we have a descending chain:Diff("M") ⊃ Diff

_{"x"}^{1}(M) ⊃ ... ⊃ Diff_{"x"}^{"n"}("M") ⊃ ...Here Diff

_{"x"}^{"n"}("M") is anormal subgroup of Diff_{"x"}^{1}("M"), which means we can look at thequotient group :Diff

_{"x"}^{1}("M")/Diff_{"x"}^{"n"}("M").Using

harmonic analysis , a real- or complex-valued function (with some sufficiently nice topological properties) on the diffeomorphism group can bedecompose d into Diff_{"x"}^{1}("M") representation-valued functions over "M".**The supply of representations**So what are the reps of Diff

_{"x"}^{1}("M")? Let's use the fact that if we have agroup 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

_{"x"}^{1}("M")/Diff_{"x"}^{"n"}("M"),we can obtain a rep of Diff

_{"x"}^{1}("M").Let's look at

:Diff

_{"x"}^{1}("M")/Diff_{"x"}^{2}("M")first. This is

isomorphic to thegeneral 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**)?:$GL^+(n,mathbb\{R\})cong\; mathbb\{R\}^+\; imes\; SL(n,mathbb\{R\})$.

We know the reps of SL("n",

**R**) are simplytensor s over "n" dimensions. How about the**R**^{+}part? That corresponds to the "density", or in other words, how the tensor rescales under thedeterminant of theJacobian of the diffeomorphism at "x". (Think of it as theconformal 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

_{"x"}^{1}("M")/Diff_{"x"}^{"n"}("M").This is a finite-dimensional group. We have the chain

:Diff

_{"x"}^{1}("M")/Diff_{"x"}^{1}("M") ⊂ ... ⊂ Diff_{"x"}^{1}("M")/Diff_{"x"}^{"n"}("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

_{"x"}^{1}("M")/Diff_{"x"}^{"m"}("M")can automatically be turned into a rep of

:Diff

_{"x"}^{1}/Diff_{"x"}^{"n"}("M")if "n" > "m". Let's say we have a rep of

:Diff

_{"x"}^{1}/Diff_{"x"}^{"p" + 2}which doesn't arise from a rep of

:Diff

_{"x"}^{1}/Diff_{"x"}^{"p" + 1}.Then, we call the

fiber bundle with that rep as thefiber (i.e. Diff_{"x"}^{1}/Diff_{"x"}^{"p" + 2}is thestructure group ) aof order "p".jet bundle Side remark: This is really the method of

induced representation s with the smaller group being Diff_{x}^{1}(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

intertwiner s between them.If the fiber is not an irreducible representation of Diff

_{"x"}^{1}("M"), then we can have a nonzero intertwiner mapping each fiber pointwise into a smallerquotient representation . Also, theexterior derivative is an intertwiner from the space ofdifferential form s to another of higher order. (Other derivatives are not, because connections aren't invariant under diffeomorphisms, though they arecovariant .) Thepartial 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.

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