In Bar-Natan's "Knots at Lunch" seminar at the University of Toronto, we are currently discussing [a talk by Alekseev at Montpellier](http://katlas.math.toronto.edu/drorbn/dbnvp/Alekseev-1006-1.php) about Rouvière's expansion of the [Duflo isomorphism](http://math.univ-lyon1.fr/~calaque/LectureNotes/LectETH.pdf) to the setting of symmetric spaces.  

We understand the definition of a symmetric space, and we know that people have written books about them; but we I don't understand in what sense symmetric spaces are useful and interesting mathematical objects. In particular:



> 
> Are there significant (analytic? geometric? algebraic?) techniques which work for symmetric spaces, but not for more general classes of homogenous spaces?
> 


What I'm trying to understand (at least vaguely) is the role of symmetric spaces in the Kashiwara-Vergne picture, and the conceptual reason one might expect Duflo's isomorphism to generalize to this specific class of mathematical objects. There must be a conceptual explanation why symmetric spaces are the natural class of objects to consider in such contexts.  

A closely related question is [THIS](https://mathoverflow.net/questions/20777/hermitian-symmetric-spaces-vs-hermitian-homogeneous-spaces).



Why symmetric spaces?
The algebra of invariant differential operators on a symmetric space is commutative, and this is certainly not true for an arbitrary homogeneous space. While it is not true that the commutativity of the algebra $D^{G}$ of $G$ invariant differential operators on a homogeneous space $X = G/H$ implies that $X$ is symmetric, if $G$ is reductive it is true that $D^{G}$ is commutative if and only if $G/H$ is weakly symmetric in the sense defined by Selberg (see E.B. Vinberg's survey in Russ. Math. Surveys 56(1)). 


