In Bar-Natan's "Knots at Lunch" seminar at the University of Toronto, we are currently discussing a talk by Alekseev at Montpellier about Rouvière's expansion of the Duflo isomorphism 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.

innerinvolution, and even simpler, the centralizer Z(t) of an involution. Brauer (I believe) proved that there are finitely many finite simple G with fixed isomorphism classes of Z(t), and proposed that finite simple groups be classified using them. Eventually, they were! $\endgroup$ – Allen Knutson Oct 21 '10 at 2:56