Consider a semisimple lie algebra $\mathfrak{g}$ and the corresponding quantum group $\mathcal U_q(\mathfrak{g})$ over $\mathbf{Q}(q)$. Consider two dominant weights $\lambda,\mu\in P^+$, a matrix entry $\phi\in L(\mu)^\ast\otimes L(\mu)$, the representation $t_{\lambda}:\mathcal U_q(\mathfrak{g})\to \text{End}(L(\lambda))$ and the product $\phi\cdot t_{\lambda}:\mathcal U_q(\mathfrak{g})\to \text{End}(L(\lambda))$ defined by $$\phi\cdot t_{\lambda}(X)=\sum_{X_{(1)},X_{(2)}}\phi(X_{(1)})t_{\lambda}(X_{(2)}),\qquad\qquad X\in \mathcal U_q(\mathfrak{g}), \triangle(X)=\sum_{X_{(1)},X_{(2)}}X_{(1)}\otimes X_{(2)}.$$ I wonder if we could say anyting about $\phi\cdot t_{\lambda}$ as a product of irreducible representations and intertwiners. Maybe to be more specific, consider a right-coideal subalgebra $B\subset U_q(\mathfrak{g})$ and a spherical function $\phi:U_q(\mathfrak{g})\to \mathbf{Q}(q)$ corresponding to $B$. Does this make the situation easier?
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