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Verlinde's fusion gives a certain "tensor product" of representations of loop groups. The category of representations of loop groups has (essentially equivalent) two incarnations. One is analytic, based on $LG=\operatorname{Map}(S^1,G)$, and one algebraic, based on $L\mathfrak g=\mathfrak g[t,t^{-1}]$. The latter of these makes sense in positive characteristic. In both cases, one constructs the "fusion" of two positive energy representations of the loop group via holomorphic induction on a thrice punctured sphere (in the analytic model, this is a disc in $\mathbb C$ minus two interior discs, and in the algebraic model, this is $\mathbb P^1(\mathbb C)$ minus three points).

Can one define a fusion product like this in positive characteristic?

I have done some searches online, but haven't even managed to figure out whether there is a reasonable category (corresponding the category of representations of positive energy in characteristic zero) of representations of positive characteristic loop groups where we could expect such a fusion product to exist.

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To answer the question in the header, there is certainly a relevant fusion rule in positive characteristic. This arises in a purely representation-theoretic context in the work of various people (Olivier Mathieu and Henning Andersen in particular). But along the way relationships have to be built among a number of representation categories in order to arrive at a transparent version of fusion rules. I'm not sure about online access, but two useful papers in Comm. Math. Physics from the early 1990s are:

H.H. Andersen, "Tensor products of quantized tilting modules" (1992)

H.H. Andersen and J. Paradowski, "Fusion categories arising from semisimple Lie algebras" (1995)

These papers arise indirectly from the influential Verlinde paper in Nuclear Physics B. Loop algebras or affine Lie algebras have representation theory in negative levels shown by Kazhdan and Lusztig to share many features with the theory for quantum groups at a root of unity based on the same type of Lie algebra. (In turn, this quantum group theory transfers in a subtle way to modular Lie algebra settings in prime characteristic.) An essential shared ingredient is the organizing role of an affine Weyl group.

In order to get a rigorous mathematical framework for "truncated" tensor products appearing in the fusion rules of Verlinde, use is made of "tilting" modules which include for small highest weights the classical-looking "Weyl modules" whose tensor products are reasonably well understood. But the tilting modules are usually more complicated, having finite filtrations with quotients which are Weyl modules (and similar filtrations involving dual Weyl modules). A key fact is that the category of tilting modules is closed under tensoring. This is the refined setting in which it makes sense to tensor, break up into a direct sum of indecomposables, and then truncate, allowing only a finite number of indecomposable objects to survive: others have "quantum dimension zero" and disappear from the picture. Technically it gets fairly complicated, but the underlying ideas are transparent from the viewpoint of representation theory.

The "tilting" modules themselves continue to be studied, but roughly speaking their formal characters are predicted by Kazhdan-Lusztig theory in the settings I indicated. (The unsolved problems are mostly in prime characteristic, where a lot is known but not everything.)

After the progress made in the early 1990s there were only a few attempts at surveys of the work for mathematicians, including one by Andersen Tilting modules for algebraic groups (1995). In any case there is substantial literature out there, which I can comment further on if it's useful.

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