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) 



$U(n)$-submodules of ${\rm SO}(2n)$-modules
you can find branching rule ${\rm SO}(2n) \to U(n)$ in the book of Knapp, Lie groups beyond an introduction, and also in some paper of his, and in the book of Zelobenko, compact Lie groups and their rep's, also try the book of Tom Dieck.


best  

jorge


