# Symmetric spaces, Horocycle spaces and intertwining operators

Let $G=KAN$ be an Iwasawa decomposition of a connected semisimple Lie group with finite center. Let us assume for simplicity that the associated symmetric space $G/K$ has rank 1. Harish-Chandras plancherel theorem gives an explicit direct integral decomposition of the left-regular representation of $G$ on $L^2(G/K)$.

$L^2(G/K) \cong \int_{i\mathbb{R}} \pi_\mu p(\mu)d\mu$.

Here, $\pi_\mu$ is the spherical principle series representation with parameter $\mu$ and $p$ is an explicitly known density function. Now one has the normalized Knapp-Stein operators $A_\mu$ from $\pi_\mu$ to $\pi_{-\mu}$. These can be pieced together to give a unitary intertwining operator on the direct integral, and hence an operator $T$ on $L^2(G/K)$. Given the prominent role of the Knapp-Stein operators in representation theory I thought that the operator $T$ must have been studied extensively but I wasn't able to find anything on this subject. Specifically I'd like to know if the operator $T$ can be given in purely geometric terms without resorting to the direct integral decomposition.

In a similiar vein, if $M$ is the centralizer of $A$ in $K$ then you can construct an analoguos operator on the space of $L^2$ functions on the horocycle space $G/MN$. Again, is there any way to define this operator in geometric terms without using the direct integral decomposition?

I'd be very happy and grateful if a person more knowledgable then I could shed some light on this and could perhaps give a few references to artciles where these operators have been studied.

This suggests that something like $Tf(g) = \int_N f(ng) dn$ is reasonable to consider on $L^2(G/K)$, insofar as it doesn't refer to the spectral decomposition. But I think it is not a bounded operator on $L^2(G/K)$.