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The very short answer is that you just replace hypercohomology with global de Rham cohomology of a D-module.

The short answer is to read Theorem 3.5 in Mirkovic-Vilonen (I am referring to the arXiv paper here) and use their definition of the functor $F$ via the weight functors $F_\nu$.

The longer answer is that you can phrase this construction in a slightly more geometric way. Suppose you choose in $G$ a Borel subgroup $B$ and denote by $T$ the quotient of $B$ by its unipotent part. Then the maps $B \to G$ and $B \to T$ induce maps on the affine grassmannians, giving a diagram $\operatorname{Gr}_G \xleftarrow{b} \operatorname{Gr}_B \xrightarrow{t} \operatorname{Gr}_T$. Recall the structure of $\operatorname{Gr}_T \cong X_*(T)$ (at least, topologically and as a group), and let's call (as in the paper) $2\rho$ the sum of the positive roots of $G$ with respect to $B$. Then that Theorem 3.5 can be understood as saying that $t_* b^! \mathcal{F}[2\rho(\lambda)]$ is a vector space (inside the derived category of vector spaces) whenever $\mathcal{F}$ is a spherical perverse sheaf (or D-module) on $\operatorname{Gr}_G$.

That is, we can define $F(\mathcal{F})$ to be the vector space $t_* b^! \mathcal{F}[2\rho(\lambda)]$ on the component $\{\lambda\}$ of $\operatorname{Gr}_T$; it is a vector space graded by $X_*(T)$ and this gives a faithful exact tensor functor from spherical D-modules to $\mathbf{Rep}({}^L T)$ (thus, a little more specific than just a fiber functor).

You might want to read the notes (written by me) from February 16th and 23rd at this seminar page.

The very short answer is that you just replace hypercohomology with global de Rham cohomology of a D-module.

The short answer is to read Theorem 3.5 in Mirkovic-Vilonen (I am referring to the arXiv paper here) and use their definition of the functor $F$ via the weight functors $F_\nu$.

The longer answer is that you can phrase this construction in a slightly more geometric way. Suppose you choose in $G$ a Borel subgroup $B$ and denote by $T$ the quotient of $B$ by its unipotent part. Then the maps $B \to G$ and $B \to T$ induce maps on the affine grassmannians, giving a diagram $\operatorname{Gr}\_G \xleftarrow{b} \operatorname{Gr}\_B \xrightarrow{t} \operatorname{Gr}\_T$. Recall the structure of $\operatorname{Gr}_T \cong X_*(T)$ (at least, topologically and as a group), and let's call (as in the paper) $2\rho$ the sum of the positive roots of $G$ with respect to $B$. Then that Theorem 3.5 can be understood as saying that $t_* b^! \mathcal{F}[2\rho(\lambda)]$ is a vector space (inside the derived category of vector spaces) whenever $\mathcal{F}$ is a spherical perverse sheaf (or D-module) on $\operatorname{Gr}_G$.

That is, we can define $F(\mathcal{F})$ to be the vector space $t_* b^! \mathcal{F}[2\rho(\lambda)]$ on the component $\{\lambda\}$ of $\operatorname{Gr}_T$; it is a vector space graded by $X_*(T)$ and this gives a faithful exact tensor functor from spherical D-modules to $\mathbf{Rep}({}^L T)$ (thus, a little more specific than just a fiber functor).

You might want to read the notes (written by me) from February 16th and 23rd at this seminar page.

The very short answer is that you just replace hypercohomology with global de Rham cohomology of a D-module.

The short answer is to read Theorem 3.5 in Mirkovic-Vilonen (I am referring to the arXiv paper here) and use their definition of the functor $F$ via the weight functors $F_\nu$.

The longer answer is that you can phrase this construction in a slightly more geometric way. Suppose you choose in $G$ a Borel subgroup $B$ and denote by $T$ the quotient of $B$ by its unipotent part. Then the maps $B \to G$ and $B \to T$ induce maps on the affine grassmannians, giving a diagram $\operatorname{Gr}_G \xleftarrow{b} \operatorname{Gr}_B \xrightarrow{t} \operatorname{Gr}_T$. Recall the structure of $\operatorname{Gr}_T \cong X_*(T)$ (at least, topologically and as a group), and let's call (as in the paper) $2\rho$ the sum of the positive roots of $G$ with respect to $B$. Then that Theorem 3.5 can be understood as saying that $t_* b^! \mathcal{F}[2\rho(\lambda)]$ is a vector space (inside the derived category of vector spaces) whenever $\mathcal{F}$ is a spherical perverse sheaf (or D-module) on $\operatorname{Gr}_G$.

That is, we can define $F(\mathcal{F})$ to be the vector space $t_* b^* \mathcal{F}[2\rho(\lambda)]$ on the component $\{\lambda\}$ of $\operatorname{Gr}_T$; it is a vector space graded by $X_*(T)$ and this gives a faithful exact tensor functor from spherical D-modules to $\mathbf{Rep}({}^L T)$ (thus, a little more specific than just a fiber functor).

You might want to read the notes (written by me) from February 16th and 23rd at this seminar page.

The very short answer is that you just replace hypercohomology with global de Rham cohomology of a D-module.

The short answer is to read Theorem 3.5 in Mirkovic-Vilonen (I am referring to the arXiv paper here) and use their definition of the functor $F$ via the weight functors $F_\nu$.

The longer answer is that you can phrase this construction in a slightly more geometric way. Suppose you choose in $G$ a Borel subgroup $B$ and denote by $T$ the quotient of $B$ by its unipotent part. Then the maps $B \to G$ and $B \to T$ induce maps on the affine grassmannians, giving a diagram $\operatorname{Gr}_G \xleftarrow{b} \operatorname{Gr}_B \xrightarrow{t} \operatorname{Gr}_T$. Recall the structure of $\operatorname{Gr}_T \cong X_*(T)$ (at least, topologically and as a group), and let's call (as in the paper) $2\rho$ the sum of the positive roots of $G$ with respect to $B$. Then that Theorem 3.5 can be understood as saying that $t_* b^! \mathcal{F}[2\rho(\lambda)]$ is a vector space (inside the derived category of vector spaces) whenever $\mathcal{F}$ is a spherical perverse sheaf (or D-module) on $\operatorname{Gr}_G$.

That is, we can define $F(\mathcal{F})$ to be the vector space $t_* b^! \mathcal{F}[2\rho(\lambda)]$ on the component $\{\lambda\}$ of $\operatorname{Gr}_T$; it is a vector space graded by $X_*(T)$ and this gives a faithful exact tensor functor from spherical D-modules to $\mathbf{Rep}({}^L T)$ (thus, a little more specific than just a fiber functor).

You might want to read the notes (written by me) from February 16th and 23rd at this seminar page.