Symmetric spaces, Horocycle spaces and intertwining operators

Question

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.

Answer 1

An explicit form is in my paper "A duality ... Advan. Math. 1970 and in my book Geometric Analysis on Symmetric Spaces AMS 2008, pp.554-556. Helgason.

Answer 2

I infer that the relevant normalizations of the intertwinings do not make them send the normalized spherical vector to itself, or else the Hilbert integral of these isomorphisms would be the identity map. A more interesting non-normalized version would be the integral over N whose meromorphic continuations gives the non-normalized intertwinings from pi(mu) to pi(-mu), whose effect on normalized spherical vectors is multiplication by an explicit constant (Gindikin-Karpelevich formula, even in general).

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 $L^2(G/K)$.

Certainly A. Knapp's book on Repn theory "by examples" discusses the individual intertwinings, but much less the spectral decomposition. Varadarajan's small book "introduction to harmonic analysis on semi-simple Lie groups" treats SL(2,R) in great detail.

Hope this is relevant to the intent of your question.