Let $\mathfrak{g}$ be a Lie algebra admitting a triangular decomposition $\frak g = N^-\oplus h\oplus N^+$ and $\gamma$ the classic Harish-Chandra isomorphism defined on the center $\frak Z$ of the universal enveloping algebra $\frak U$ of $\frak g$.
On the other hand, for a symmetric pair $(\frak g, k)$, we have the Harish-Chandra homomorphism $\Gamma$ on $\frak U^k$ (centralizer of $\frak k$ in $\frak U$) w.r.t. the Iwasawa decomposition $\frak g = n\oplus a\oplus k$. Specifically, it's the (restricted Weyl vector $\rho$-shifted) projection w.r.t. $\frak U = (nU+Uk)\oplus S(a)$ restricted to $\frak U^k$. (Note this $\gamma$ is the specialization of $\Gamma$ for $(\frak g\oplus g, g)$.)
I believe it's a standard result that these two maps are related via the following commutative diagram: $\require{AMScd}$ \begin{CD} \mathfrak{Z} @>>> \mathfrak{U^k}\\ @V \gamma V V @VV \Gamma V\\ \mathfrak{S(h)} @>>> \mathfrak{S(a)} \end{CD} Here we assume that $\frak h =a+t$ is a Cartan subalgebra extended from $\frak a$. The bottom map is just a projection induced from $\frak a+t \rightarrow a$.
What might be a good reference for this statement? I've seen Knop's paper https://doi.org/10.2307/2118600 confirming my guess but it is completely beyond my scope (Knop deals with varieties with a $G$-action). This above statement is however purely "Lie algebraic" and shouldn't involve too much rep theoretical machinery to prove.
I'd really appreciate any useful suggestions and references on this "theorem".
