If $\lambda$ is a HarishChandra paramater. Let $\pi_\lambda$ it's associated discrete series, it's known by the minimal Ktype thm that every Ktype of $\pi_\lambda\mid_K$ has highest weight of the form $\lambda+\rho_n\rho_c+C$ with C=sum of noncompact positive roots. Even when $\lambda+\rho_n\rho_c+C$ isn't a highest weight of $\pi_\lambda\mid_K$, we can say that $\lambda+\rho_n\rho_c+C$ is dominant with respect to the system positive of K?
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for the split form of G_2, consider the system of positive roots so that the short root is compact and the long root is non compact, call it b, then for large n, $\lambda + \rho_n \rho_c +nb $ is not dominant for the Kchamber that contains $\lambda. $ 

