# How to show the compatibility between Duflo isomorphism and Harish-Chandra isomorphism for semi-simple Lie algebras?

I was told that the Duflo isomorphism is compatible with the Harish-Chandra isomorphism when the Lie algebra $\mathfrak{g}$ is semi-simple. However I cannot see why this is true. All I can show is that Harish-Chandra map $\tilde{\gamma}: Z(U(\mathfrak{g}))\to S(\mathfrak{h})^{\mathcal{W}}$ combined with the inverse of the Chevalley isomorphism $s: S(\mathfrak{g})^{\mathfrak{g}}\to S(\mathfrak{h})^{\mathcal{W}}$ gives an algebra isomorphism $U(\mathfrak{g})^{\mathfrak{g}}\to S(\mathfrak{g})^{\mathfrak{g}}$, but how can one prove that this is exactly the Duflo isomorphism?

• I guess "compatible" is a key word here, since there isn't usually an obvious way to get an algebra isomorphism from the center of the enveloping algebra to fixed points on the symmetric algebra. Duflo's basic idea is to twist a natural map by an operator which in the semisimple case corresponds to the weight $-\rho$. But getting $\rho$ requires some transition from Duflo's general picture expressed by differential geometry. I'm not sure what is the best method, since Dixmier's book was published earlier than Duflo's 1977 paper: ams.org/mathscinet-getitem?mr=0444841 Apr 21, 2015 at 22:47
• @JimHumphreys Thank you very much for the reply. Yes, what I am trying to see is why the idea of Duflo in his 1977 paper (which is obtained by composite an infinite dimensional operator to the vector space isomorphism by PBW theorem) coincides with the twisted map of Harish-Chandra in the semi-simple case. The only reference I can find regarding Duflo's original proof is his French paper, which is rather a challenge. I was hoping that someone might be able to explain his idea to me so that I can see how those two isomorphisms coincide in the semi-simple case.
– Xuxu
Apr 22, 2015 at 4:47
• Duflo approaches all of this rather analytically (and it looks complicated), but his basic idea in the semisimple case might be seen a little more clearly in the last section of his earlier paper (in English) published in 1975 but based on his conference lecture in 1971 (which may be hard to track down in libraries): ams.org/mathscinet-getitem?mr=0399194 Apr 22, 2015 at 14:26