Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$, and $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ the unique algebra morphism such that $\Psi_a(\phi) \circ P_a = P_a \circ \phi$ for all $\phi \in \mathcal{U}_q$. We study $\Psi_a(E)$ in this question. 21a. Determine $\Psi_a(E)(v_0)$. 21b. Deduce $\Psi_a(E^{\ell})$. 21c. Calculate the dimension of the vector subspace $\mathbf{C}[\Psi_a(E)]$. 21d. Calculate the eigenvectors of $\Psi_a(E)$.
Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$, and $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ the unique algebra morphism such that $\Psi_a(\phi) \circ P_a = P_a \circ \phi$ for all $\phi \in \mathcal{U}_q$. We study $\Psi_a(E)$ in this question.
21a. Determine $\Psi_a(E)(v_0)$.
21b. Deduce $\Psi_a(E^{\ell})$.
21c. Calculate the dimension of the vector subspace $\mathbf{C}[\Psi_a(E)]$.
21d. Calculate the eigenvectors of $\Psi_a(E)$.