Some time ago I was told there's an interesting classical Satake correspondence which I will write as 

$$[\mathop{\mathrm{disk}} \Rightarrow G] \,\backslash\, [\mathop{\mathrm{disk}^\times} \Rightarrow G] \,/\, [\mathop{\mathrm{disk}} \Rightarrow G] \,=\, X_*/W \,=\, G^\vee\mbox{-reps}$$

(where $G$ is reductive, $\mathrm{disk}$ could be the spectrum of either $\mathbb Z_p$ or $k[[t]]$ and $\Rightarrow $ denotes algebraic morphism) and its categorified/geometric version (equivariant perverse sheaves on affine grassmanian of $G$ form the tensor category of representations of $G^\vee$).

I think I'm missing the larger context here, though. I don't mean the context of perverse sheaves, geometric Langlands, etc. On the contrary, I feel like I miss any intuition for classical representation theory. Why would statements like this be interesting?

I wasn't able to find anything in [wikipedia](http://en.wikipedia.org/wiki/Satake_correspondence) or [nLab](http://ncatlab.org/nlab/search?_form_key=4209aaac3878c2dab6d077f12cd5c0363e94a24c&query=satake+correspondence).

One thing I know is that the correspondence allows us to construct the Langlands dual group in a natural way. But still, it would be interesting to know if it's a part of larger picture and if  there are related results. 

> **Question:** is there an intuition for Satake correspondence that would make its statement obvious?