Introduction to Mathematical Logic, Sixth Edition
《Introduction to Mathematical Logic, Sixth Edition》 · 第 585 页 · Chapter 12，Section 12.3.3「Spherical Representations」，Theorem 12.3.13
We now determine the irreducible representations having a $K$-fixed vector. [...] Theorem 12.3.13. Let $(\pi^\lambda, V^\lambda)$ be an irreducible regular representation of $G$ with highest weight $\lambda$ (relative to $B$). The following are equivalent: 1. $V^\lambda$ contains a nonzero vector fixed by $K$. 2. $V^\lambda$ contains a nonzero vector fixed by $MN^+$. 3. $t^\lambda = 1$ for all $t \in T = H \cap K$.