Is the Nelson-Laplacian $\sum_{i=1}^{n}X_{i}^{2}$ on a Lie group $G$ an elliptic differential operator?

Question

Let $G$ be a Lie group of dimension $n\in \mathbb N$ and let $X_{1},...,X_{n}$ be a basis of the Lie algebra $\mathfrak{g}$ of $G$. Let $\mathfrak{g}$ act by left-invariant vectorfields on smooth functions on $G$. Is it true that the left-invariant differential operator given by the element $$ \sum_{i=1}^{n}X_{i}^{2} $$ in the universal enveloping algebra $U(\mathfrak{g})$ is an elliptic differential operator ? If so how can one prove this?

Im currently working with the book Representation theory of semisimple groups by Anthony W. Knapp and im currently stuck in the proof of Theorem 8.7 where he proves that all matrix coefficients $g \mapsto \langle \pi(g)v,w\rangle$ are real analytic functions whenever $(\pi,V)$ is an admissible representation of G on a Hilbertspace V and $v$ is a K-finite vector in $V$ and $w \in V$. Here G is assumed to be connected linear reductive. He shows that a certain elliptic differential operator $D$ annihilates the matrix coefficient and since it is elliptic and has real analytic coefficients the matrix coefficient has to be analytic aswell. If I would know that $\sum_{i=1}^{n}X_{i}^{2}$ is elliptic I could deduce that $D$ is elliptic as well.

Knapp says that one can see that it is elliptic if one looks at $\sum_{i=1}^{n}X_{i}^{2}$ in canonical coordinates of the first kind given by the exponential map relative to the basis $X_{1},...,X_{n}$ . I tried this but I failed. I would really appreciate the help. Cheers

Answer 1

I'm not entirely confident that I remember the context of such a discussion in Knapp's books, but...

First, the literal expression $\sum_i X_i^2$ for an arbitrary basis of the Lie algebra is not well-defined, by far. Not even close.

Second, a well-defined thing is $\sum_i X_i\cdot X_i^*$, where $X_i^*$ are the dual basis vectors via the Killing form... for a semi-simple Lie algebra. This gives the Casimir operator, whose basis-independence can be verified by silly calculations (as are popular), or by coordinate-free characterizations.

Even then, the Casimir element is not at all elliptic unless the Lie group is compact. But, on functions on $G$ with restricted right $K$-type, it is elliptic, etc.