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We give an explicit computation of the ring structure in wrapped Floer homology of real Lagrangians in $A_k$-type Milnor fibers. In the $A_k$-type plumbing description, those Lagrangians correspond to cotangent fibers or diagonal Lagrangians. The main ingredient of the computation is an idea of the Seidel representation. For a technical reason, we first carry out computations in v-shaped wrapped Floer homology, and this in turn gives the desired ring structure via the Viterbo transfer map.

A computation of the ring structure in wrapped Floer homology