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.



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> 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?
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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).



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


