Let $G$ be a semisimple real Lie group. Let $\Delta \in U(\mathfrak g_{\mathbb C})$ be the Casimir element associated to the Killing form on the complexified Lie algebra $\mathfrak g_{\mathbb C}$ of $G$.

On the Wikipedia article on the Casimir element, it says:

"the choice of an invariant bilinear form on $\mathfrak g_{\mathbb C}$ corresponds to a choice of bi-invariant Riemannian metric on $G$. Then under the identification of the universal enveloping algebra with the left invariant differential operators on $G$, the Casimir element maps to the Laplacian of $G$ (with respect to the corresponding bi-invariant metric)."

What is the Riemannian metric on $G$ associated to the Killing form? How is it defined? And what is meant by the Laplacian of $G$ with respect to a Riemannian metric? There seem to be several inequivalent notions of a Laplacian operator.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.