On the Kashiwara–Vergne conjecture

Abstract

Let $G$ be a connected Lie group, with Lie algebra $g$. In 1977, Duflo constructed a homomorphism of $g$-modules $Duf: S(g) -> U(g)$, which restricts to an algebra isomorphism on invariants. Kashiwara and Vergne (1978) proposed a conjecture on the Campbell-Hausdorff series, which (among other things) extends the Duflo theorem to germs of bi-invariant distributions on the Lie group $G$. The main results of the present paper are as follows. (1) Using a recent result of Torossian (2002), we establish the Kashiwara-Vergne conjecture for any Lie group $G$. (2) We give a reformulation of the Kashiwara-Vergne property in terms of Lie algebra cohomology. As a direct corollary, one obtains the algebra isomorphism $H(g,S(g)) -> H(g,U(g))$, as well as a more general statement for distributions.