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I'm looking for a reference that will set me straight on a few things.

It started out with densities. In John Lee's book, "Introduction to Smooth Manifolds", densities on vector spaces are functions that satisfy $$\mu(Tv_1,\dotsc,Tv_n)=|\det\; T\,|\;\mu(v_1,\dotsc,v_n)$$ for $T:V\to V$. They're like volume forms composed with absolute values. Densities are defined on manifolds and transform the obvious way under changes of coordinates. They're nice because you can integrate them in the absence of an orientation. And they're natural because the structure group is an irreducible representation of $\operatorname{GL}(n)$.

This representation is a little bit unfamiliar in that it is not constructed out of tensor products of the fundamental rep and its dual. One of the reasons that symmetric and antisymmetric tensors on a manifold are important is because their structure groups belong to irreducible representations of $\operatorname{GL}(n)$. Googling for more information on the subject let me to an old USENET thread where I learned among other things that things that pick up a sign under negative determinant change of basis are called pseudoforms, not densities. Any (symmetric, antisymmetric, form, vector, etc.) can be turned "pseudo" by tensoring it with the determinant bundle. Densities are things with an exponent in their change of basis formulas, and these also form an irrep of $\operatorname{GL}(n)$. Jet bundles are another example of a bundle whose structure group is not a tensor product

I'm left with some questions

  1. What are densities? Are they what Lee says, or what the USENET guys say? Are there differing conventions? If you use the word "density" for Lee's definition, then what do you call the thing with the exponent?
  2. Are there other nontensorial reps than density and pseudoforms? What about jet bundles?
  3. What about a rep of the universal cover of $\operatorname{GL}(n)$? I know that for $\operatorname{SO}(n)$ this gives us spinor reps. What about $\operatorname{GL}(n)$?
  4. And what about $\operatorname{SO}(n)$ and subgroups of $\operatorname{GL}(n)$?
  5. I would like a good reference text that covers this material in depth. Especially as it relates to differential topology on smooth manifolds. I did find some a book on the representation theory of Lie groups which proved the irreps of $\operatorname{GL}(n,\mathbb{C})$, but it was difficult to understand.
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2 Answers

An elegant definition of densities follows from a nice piece of linear algebra invented by Mumford , which can be found in a footnote (!) to page 88 of his book Algebraic geometry I .
(There is no volume II: as someone remarked, Hell is paved with volumes I...).

Consider a real vector space $V$ of dimension 1.
Mumford associates to it another vector space $|V|$ of dimension 1 whose vectors are symbols $r|v|$ where $r\in \mathbb R$ and $v\in V$.
Equality of vectors follows from the axiom
$$s|rv|=s|r||v| \quad (*)$$

in which $r,s \in \mathbb R, \; v\in V \; $ and where of course $|r| $ is the absolute value of the real number $r$.
For any $v\neq 0\in V$, the element $|v|\in |V|$ is a basis of $|V|$.

Of course if you want to set this up rigorously, you shouldn't talk of symbols, but take the free vector space with basis $V$ and divide out by the hyperplane created by the relations$(*)$

The volume elements of an $n$-dimensional vector space $E$ are the elements of Mumford's $|\wedge^n E^*$| associated to the top exterior product $\wedge^n E^*$ of the dual of $E$.
Since all this is completely canonical, it extends to vector bundles on an $n$-dimensional manifold $M$ and you can define densities as sections of the line bundle $|\mathbb \Omega^n(M)|$ associated to the line bundle $\mathbb \Omega^n(M)$ of differential forms of maximal degree .

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So Mumford agrees with Lee: things that are like absolute values of forms are called densities, not pseudoforms. But how to understand all these gadgets as representations of GL(n)? –  Joe Hannon Feb 9 '12 at 21:47
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I found this resource on the MIT OWC: http://ocw.mit.edu/courses/mathematics/18-712-introduction-to-representation-theory-fall-2010/lecture-notes/MIT18_712F10_ch4.pdf

It contains a theorem which states that all irreducible polynomial representations of $\text{GL}(n)$ are of the form of a symmetrized tensor rep tensored with a power of the determinant representation. According to Wikipedia's definition, but not Lee's, that power of the determinant makes this is a density.

This is progress toward what I'm looking for. I believe I could study this resource and learn about the polynomial representations of $\text{GL}(n).$ But I guess the $|x|$ is not a polynomial in $x$, so the "pseudo" reps are not to be found here. That's an important part of my question.

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