TITLE: Langlands duality and multiplying cocharacters
QUESTION [13 upvotes]: 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.

REPLY [6 votes]: This is another version of Reimundo Heluani's answer, avoiding perverse sheaves.
In general, the $H$-orbits on $H/P \times H/Q$ are indexed by $W_P \backslash W_H/W_Q$. If $H = {}^L G(\mathcal K)$, so $W_H$ is the affine Weyl group $W_G \ltimes $[the weight lattice], and $P=Q={}^LG(\mathcal O)$, then $W_P \backslash W_H/W_Q =$ dominant weights. For $L,L' \in \mathcal Gr$, write $L \stackrel{\lambda}{\to} L'$ if $(L,L')$ is in the $\lambda$-orbit, for $\lambda$ a dominant weight.
Now consider the space of "triangles"
$$\mathcal V_{\lambda\mu\nu} := \{(L,L') : L,L' \in \mathcal Gr, base \stackrel{\lambda}{\to}
L \stackrel{\mu}{\to}
L' \stackrel{\nu}{\to}
base \}$$
and I believe the statement is that $V_{\lambda\mu\nu} \cong H_{top}(\mathcal V_{\lambda\mu\nu})$.