If $\Bbb H$ denotes the quaternions one can write every element $a+bi+cj+dk\in \Bbb H$ as $1(a+bi)+j(c-di)$, so $\{1,j\}$ is a $\Bbb C$-basis of $\Bbb H$.

Now let $w_1,\ldots, w_n$ be the standard basis of $\Bbb H^n$ and write $w_a=z_a+jz_{n+a}$ as above. Now let $G=Sp(n,1)$ be the group of all $\Bbb H$-linear transformations of $\Bbb H^{n+1}$ preserving the quadratic form $|x_1|^2+\ldots+|x_n|^2-|x_{n+1}|^2$ and $K$ be the maximal compact subgroup of $G$.

Finally for $X\in\mathfrak{k}$ in the Lie algebra of $K$ and $f:K\rightarrow \Bbb C$ set $(\rho(X)f)(y):=\frac{d}{dt}|_{t=0}f(y\exp(tX))$. Now my question is: How is $\rho(X)$ a differential operator of $\Bbb H^n$? This is claimed in Wallach on page 143, but I dont see how this can work.

Maybe it is helpful that I know a $G$-action on the unit ball $D^n$ in $\Bbb H^n$ where the stabilizer of $0$ is $K$.


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