We have the Chevalley restriction theorem:
The restriction of the natural map from the ring of polynomial class functions on the lie algebra to the ring of polynomial functions on is injective, and its image coincides with the invariant polynomial ring with respect to the action of the Weyl group on .
Note that , so , the ring of polynomial functions on , is identified with , which is the ring of polynomial functions on .
We further have an isomorphism of algebras from to the algebra of polynomial class functions on by taking character.
The Bruhat decomposition tells us that the double coset algebra is isomorphic to (This is the specialization of the one-parameter family of Iwahori-Hecke algebras specialized at ; for all other values of except at roots of unity, Tits prove it is isomorphic to it is also isomorphic to ; See here, point 4). We would like to obtain similar description for maximal compact . See the wiki article. This will give us the unramified local Langlands correspondence, where the parameter is just a point of , or a semisimple element of up to conjugacy. see here for a sketch of the proof. The key is Iwasawa’s decomposition .
Proof that is -invariant (note that it is important to put in the factor (see page 147 of Cartier’s article).
Reference:
https://virtualmath1.stanford.edu/~conrad/JLseminar/Notes/L4.pdf
https://kentajsuzuki.github.io/seminar-talk/yuta-talk.pdf
https://people.math.harvard.edu/~gross/preprints/sat.pdf