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The Geometric Satake correspondence (due to Lusztig, Ginzburg, Mirkovic-Vilonen) relates perverse sheaves on the Loop Group $\hat{G}$ (with their convolution product) to the Representations of the Langlands dual Group $G^\vee$ (with the usual tensor product). More precisely, it can be stated as an equivalence of the above two categories. Now, in most presentations, the category of representations appearing on the dual side is that of finite dimensional representations. So, let me restrict my attention to that.

I would like to know if there are known results/conjectures about how this correspondence is expected to behave under a restriction map on the $\hat{G}$ side to subgroups (proper Levis or more generally, pseudo-Levis). Specifically, I would like to know if there is some analog of the conjectures of Gan-Gross-Prasad (which is stated in the setting of infinite dimensional representations and their associated Langlands parameters) in the Geometric Satake setting that I recalled above.

There is this earlier question which has a partial answer to what I am asking. But, it is not obvious to me if this readily leads to something like Gan-Gross-Prasad.

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