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More specifically, let $S_k$ be the spin representation of $\mathrm{Spin}(2k+1)$. Then is there are action of $\mathrm{Sp}(2r-2)$ on $\otimes^{2r}S_k$ which commutes with the action of $\mathrm{Spin}(2k+1)$?

I am not asking for a duality. That is, I do not require that each isotypic vector space is an irreducible representation of $\mathrm{Sp}(2r-2)$.

The motivation is that the space of invariant tensors in $\otimes^{2r}S_k$ looks like the irreducible representation of $\mathrm{Sp}(2r-2)$ with highest weight $k\omega_{r-1}$. For instance, for $k=1$, both vector spaces have dimension given by the Catalan numbers.

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