# Harish-Chandra isomorphism for compact symmetric spaces

I would be interested to have an explicit description of the algebra of invariant differential operators on functions on a compact symmetric space $G/K$. A reference would be especially useful.

Below I am making it more precise what kind of description I need. It is a generalization of the famous Harish-Chandra isomorphism which I recall first.

Let $\mathfrak{g}$ be a complex reductive Lie algebra. Fix a Cartan subalgebra $\frak{h}$ and its positive roots. Then there exists an algebra isomorphism

$$\gamma: Z(U(\mathfrak{g}))\mathrel{\tilde{\to}} Sym(\mathfrak{h})^W$$

of the center of the universal enveloping algebra $Z(U(\mathfrak{g}))$ and the algebra $Sym(\mathfrak{h})^W$ of $W$-invariant polynomials on $\mathfrak{h}^*$ where $W$ is the Weyl group, which is uniquely characterized by the property that any element $x\in Z(U(\mathfrak{g}))$ acts on an irreducible representation $V_\lambda$ with highest weight $\lambda\in \mathfrak{h}^*$ by the scalar $\gamma(x)(\lambda+\rho)$ where $\rho$ is the half sum of positive roots.

The algebra $Z(U(\mathfrak{g}))$ can alternatively be viewed as the algebra of bi-invariant differential operators on functions on a Lie group $G$ such that $\mathfrak{g}$ is the complexification of $Lie(G)$, or the algebra of invariant differential operators on the symmetric space $(G\times G)/G$ where $G\subset G\times G$ is the diagonal subgroup.

I have heard that there exists a generalization of the above isomorphism, but in the context of compact symmetric spaces (the relevant description for symmetric spaces of non-compact type is contained in Helgason's book "Groups and Geometric Analysis"). I would be happy to have a precise statement and a reference.

Let $(G,K)$ be a symmetric pair of compact type. Assume that $G$ is connected, but $K$ does not have to be (it is the standard convention in the literature, though I will need the case of disconnected $K=O(n)$). I have heard that the algebra of invariant differential operators on $G/K$ is isomorphic to the algebra of $W'$-invariant polynomial on the Cartan space of the symmetric space $G/K$, where $W'$ is the Weyl group of $G/K$. Moreover this isomorphism is somehow uniquely characterized by the action of the differential operator on all irreducible subspaces of $L^2(G/K)$ in terms of their highest weights (all of them are known exactly for a compact symmetric space $G/K$).

• I made a few minor edits but am not close enough to the subject to provide a solid answer. However, you may want to look at Chapter VII in Helgason's 1978 book Differential Geometry, Lie Groups, and Symmetric Spaces (now an AMS reprint), which explicitly discusses compact symmetric spaces. However, this (like probably most treatments) assumes that $K$ is connected. Aside from that, my impression is that the compact spaces involve only parts of classical finite dimensional representation theory and related differential operators. Sep 11, 2014 at 13:34
• I should add that any formulation here must involve some careful bookkeeping with roots and Weyl groups for $G$ or "restricted" versions for $K$. The notation, as in Helgason's chapter, gets heavy at times, and varies somewhat in other sources. The relevant finite dimensional representations here correlate with appropriate parabolic (generalized) Verma modules, as in the paper by Lepowsky cited by Francois. Sep 12, 2014 at 20:08

I believe the isomorphism you want is Theorem 8.2 of Lepowsky's Generalized Verma modules, the Cartan-Helgason theorem, and the Harish-Chandra homomorphism (1977). Not sure about a characterization in terms of highest weights in $L^2(G/K)$...