Let $\Gamma_{(\lambda_1, \dots, \lambda_{n})}$ denote an irreducible $SO(2n)$-module with highest weight $(\lambda_1, \dots, \lambda_n)$ and let more specifically $X = \Gamma_{(2\lambda, \dots, 0)}$ and $Y = \Gamma_{(2\lambda_1, \dots, 2\lambda_n)}$, where at least one of the $\lambda_j$ with $j>1$ is not equal to $0$. Furthermore, let $W \leq X$ be a (irreducible) $U(n)$-submodule of $X$, and $\tilde{W} \leq Y$ a (irreducible) $U(n)$-submodule of $Y$. My Questions:

1) Is it true that for any $X,Y,W,\tilde{W}$ choosen in such a fashion \begin{align*} W \ncong \tilde{W}? \end{align*}

2) Is there a branching rule for $U(n)$-submodules of $SO(2n)$-modules? (This obviously relates to Question 1)