# Half Haunted: The 1/2 in Harish-Chandra via the Fourier Transform

This post is written together *with Josh Mundinger*. Last time, we compared the Harish-Chandra isomorphism \(Z(U\mathfrak g) \cong (\text{Sym} \mathfrak h)^{W,\cdot}\) for \(\mathfrak g= \mathfrak{sl}_2\) to the Duflo isomorphism \(Z(U\mathfrak g) \cong (\text{Sym } \mathfrak g)^{\mathfrak g} \cong (\text{Sym} \mathfrak h)^W\), and found that they differ exactly by a translation by \(\rho\). In this blog post, we study just the Harish-Chandra map \(Z(U\mathfrak g) \to \mathbb C[\mathfrak h]\), using the Fourier transform to explain why the image is invariant under the dot action \((W,\cdot)\). Recall that the Harish-Chandra map sends \(z \in Z(U\mathfrak g)\) to the action of \(z\) on the Verma module \(M_\lambda\). The dot action of \(W\) is defined by \(w \cdot \lambda = w(\lambda + \rho) - \rho\). Thus,
for \(\mathfrak sl_2\), we need to show that the center of \(U\mathfrak sl_2\) acts by the same scalar on \(M_{\lambda}\)and \(M_{-\lambda - 2}\).