Recall that there is a bijection between irreducible representations of a compact real Lie group $G$ and the cocharacters (homomorphisms $U(1) \to G$, modulo conjugation) of the Langlands dual group $^LG$.

The irreducible representations of $G$ have additional structure related to tensoring representations: Given representations $\alpha, \beta, \gamma$ of $G$, we have the invariant subspace $V_{\alpha\beta\gamma}$ of the tensor product $\alpha\otimes\beta\otimes\gamma$. (Or, if you prefer, the space $V_{\alpha\beta}^{\gamma^*}$ of homomorphisms from $\gamma^*$ tp $\alpha\otimes\beta$.)

Is there an intrinsic way to define a Langlands dual structure on the cocharacters of $^LG$? In other words, in a natural way (and without using Langlands duality) associate to cocharacters $a,b,c$ of $^LG$ a vector space $V_{abc}$?

One possibility would be to think of the cocharacters as (equivalence classes of) geodesics in $^LG$ and replace the tensoring of $G$ representations with the splicing of $^LG$ geodesics. The resulting families of broken geodesics $a \cdot b$ could flow to actual geodesics $\{c_i\}$. Before pursuing this idea I wanted to check whether there are already known answers to the main question above.

ADDED LATER:

The motivation for the above question was geometric Langlands TQFTs applied to the operation of gluing two disks together to obtain another disk. I had forgotten that on the Rep($G$) side a 2-sphere gives the same monoidal category as a disk, and that therefore the geometric Satake isomorphism answers my question. I still wonder whether thinking in terms of disks instead of spheres would give a different (but presumably equivalent) construction of the monoidal product on the cocharacter side.