Abel transform is an * isomorphism for SL(2, R)

Question

Assume we conisder $G= SL(2, R)$, $K=SO(2)$ and $N$ the strict upper triangular matrices in $G$, $A$ diagonal matrices, and the Borel supgroup $B=NA$, $W$ Weyl group.

Then we have an isomorphism of $*$ algebras $C_c(G//K)$ ($K$ bi invariant functions) and $C_c(A)^W$, see e.g. Lang "$G$";) page 70.

The $*$ isomorphism is given by $$ H\phi(n) = \Delta_B(m)^{-1/2} \int_N \phi(mn) \mathrm{d} n.$$

Lang proves this by explicitely constructing the inverse.

Is there a way to prove the injectivity and surjectivity of this map without explicit computation of the inverse, but rather by using the Iwasawa and Cartan decomposition directly?

Feel free to argue for any other Lie group or reductive group, but I really would like to stay on the group level rather than working with Lie algebras (which I am not incredibly comforatble with).

Answer 1

There is a direct proof of injectivity, using representation theory, due to R. Godement, A theory of spherical functions I, Trans. Amer. Math. Soc. 73 (1952), 496-556.

You may also have a look at my old paper "A global approach to spherical functions on rank 1 symmetric spaces", Nieuw Archief voor Wiskunde, 5 (1987), no. 1, 33–52, which is also motivated by the desire of staying on the group level, rather than using infinitesimal methods. In retrospect, I would say that it is worth investing a bit in Lie algebras!