# Laplacian type operator on compact Lie group

Consider the operator $S = \sum_{i,j} X_{ij}^2$ on $L^2(SO(n+1))$, where $X_{ij}$ generates the rotation of the sphere $S^n$ in the $ij$-plane keeping the $(n - 2)$ complementary directions fixed. Consider the operator $-S + X^2 - i\lambda X$, where $X$ is any one of the $X_{ij}$'s. I want to see when this operator is positive semi-definite (meaning, for what range of $\lambda$). I think there is a subelliptic estimate for the operator $-S + X^2$, but it is the $\lambda X$ part that is bothering me.

It might be possible that an answer is only possible for certain values of $n$. Any hints would be well-appreciated. Also, could someone point out a source where I can read about such problems?

Since $X$ commutes with the Laplacian on $\mathrm{SO}(n{+}1)$, which is $-S$, it must preserve the eigenspaces of $-S$ on $L^2$, which are given by the matrix coefficients of the irreducible representations of $\mathrm{SO}(n{+}1)$, by the Peter-Weyl Theorem. You just need to know the (imaginary) eigenvalues of $X$ in each of these irreducible representations (and the eigenvalue of $-S$ on that representation), and this will answer your question. This is a routine computation. Look in any book on the representation theory of Lie groups, such as Knapp's "Lie groups: Beyond an introduction". – Robert Bryant May 14 '13 at 19:35
@Robert Bryant Thanks a lot for your comment. I have been reading the book by Knapp, but I don't think I have yet reached the point where such things are discussed (it is a big book after all). Can you please point out which chapter is relevant, so I can read backwards and pick up the necessary prerequisites as and when required? Also, if in the above question, I try to replace $SO$ by a compact Lie group $G$, is there an answer for that? – heychhanno Jun 9 '13 at 16:45