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Suppose $G$ is a locally compact group, $X$ is a locally compact space equipped with a nice right action $r_g$ of $G$ (I suspect every orbit being compact qualifies as "nice" here, but this probably isn't the greatest generality possible), and $(V,\rho)$ is a finite-dimensional left representation of $G$. Is there any characterization, in terms of some kind of data on the orbit space, of locally-defined $V$-valued measures on $X$ (in the sense of a system of measures on every compact subset of $X$ that behave well under restriction) which respect $G$'s action in the sense that $\left(\left(r_g\right)_*\mu\right)(E)=\rho_g\mu(E)$ for every precompact Borel set $E$? What if we let $V$ be a Banach space or similar? In the case where $X=G$ and $V$ is trivial, these are just Haar measures, but I've been unable to find any discussion of this sort of problem in greater generality.

(Here, all spaces are Hausdorff and all measures are Radon.)

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  • $\begingroup$ Perhaps you should take a look at the concept of Haar systems on groupoids. It is not exactly what you are looking for but it is a central concept for many generalizations of group actions and it might be related to what you want. $\endgroup$ – Ruy Dec 26 '20 at 20:19
  • $\begingroup$ @Ruy Yes, I think this or something like this is closely related. Thank you. $\endgroup$ – Curious Dec 27 '20 at 21:21
  • $\begingroup$ (However, I think this notion has the disadvantage of requiring a kind of nonsingularity, e.g. a Haar measure supported on a single orbit is too singular unless that orbit is isolated.) If it helps, I only care about the case where G and X are specific second-countable manifolds over a local field and all relevant morphisms are analytic. I think something like this case isn't too hard to characterize using some kind of stratification. $\endgroup$ – Curious Dec 27 '20 at 21:35
  • $\begingroup$ I hadn't read Bourbaki, but skimming it appears Bourbaki covers the case where the action is proper and the character is one-dimensional. The generalization of the character, I think, is not too hard; but can the assumption that the action be proper be weakened to the quotient being Hausdorff? This seems much harder. $\endgroup$ – Curious Dec 27 '20 at 22:10
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The answer seems a bit silly, which explains why I couldn't find any discussion. Some of what I say may not be strictly correct, but I think it is qualitatively correct. Suppose every orbit is closed in $X$, and suppose $X/G$ is also LCH.

One can construct invariant measures as follows. Take a measure $\mu$ on $X/G$, and take measures $\mu_O$ on every orbit $O$ of $X$ such that for every $f\in C_c(X)$, the function $O\mapsto\int_Of\,d\mu_O$ is in $L^1(\mu)$. Then the Radon measure $\nu$ defined by $\int f\,d\nu=\int_{X/G}\int_Of\,d\mu_O\,d\mu$ is invariant if and only if $\mu$-a.e. $\mu_O$ is invariant.

I think the converse, that every positive invariant measure arises this way, can be argued (probably assuming second-countability) by disintegration and extension theorems, though I'm not up to working through the details at this particular moment. This reduces the untwisted case to studying the homogeneous case, which can be classified in terms of modular functions; see Loomis' book, pages 132-133, or Bourbaki's section on homogeneous spaces.

The twisted case, I think, is a straightforward adaption of these ideas with a little more finagling involving stabilizers; I'll think about this in the morning. You may need to assume the representation is unitary.

If $X/G$ is not Hausdorff, we might have no good invariant measure. For example, if $K$ is a local field, the multiplicative action of $K^*$ on $K$ has a two-point non-Hausdorff quotient space. A standard fact of number theory in this case is if the representation $\rho$ is sufficiently bad, the twisted-invariant measure is not finite near zero, so you cannot integrate normally. One can still define a kind of twisted-invariant integral (this is the local theory in Tate's thesis), but in most modern treatments it lives in a much sketchier space of distributions.

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