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Let $K$ be a compact connected Lie group, wiewed as subgroup of unipotent matrices. Let $G=\mathfrak{k}^\mathbb C$ be the complexification with Lie algebra $\mathfrak{g}=\mathfrak{k}\oplus i\mathfrak{k}$.

Let $(\Phi_\lambda, V)$ be an irreducible finite dimensional representation of $K$ on complex vector space $V$, with highest weight vector $\lambda$. V admits an inner product such that $K$ acts as isometry, i.e. $$ \langle \Phi_\lambda(k)v_1,\Phi_\lambda(k)v_2\rangle=\langle v_1,v_2 \rangle,\quad v_i\in V,k\in K $$

We can extend the representation $(\Phi_\lambda,V)$ of $K$ to be a holomorphic representation of $G=K^\mathbb C$. Recall for $g\in G$, $g=k\exp(H)$ with $k\in K$ and $H\in i\mathfrak{k}$.

So what is the action of $\Phi_\lambda (g)$ with respect to the inner product? $$ \langle\Phi_\lambda(g)v_1,v_2\rangle=? $$

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