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 $\mathrm{Hom}$) 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 or nLab.

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

Question: is there an intuition for Satake correspondence that would make it clearits statement obvious?

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 $\mathrm{Hom}$) 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 or nLab.

One thing I know is that the correspondence allows us to construct the Langlands dual group in a natural way. But still, is it a part of larger picture? Are there related results? Is there an intuition for Satake correspondence that would make it clear?