Let $X_1, X_2, \ldots, X_m \sim N(\mu_1, \Sigma)$ and $Y_1, Y_2, \ldots, Y_n \sim N(\mu_2, \Sigma)$.
(Here, $\Sigma$ is the variance-covariance matrix of the 2 multivariate Normal distributions stated above; $X_1, \ldots, X_m, Y_1, \ldots, Y_n, \mu_1 \; \textrm{and} \; \mu_2$ - all being vectors.)
Now for some $\alpha$ (which is a vector), we compute : $$\alpha^{'}X_1, \alpha^{'}X_2, \ldots, \alpha^{'}X_m$$ and $$\alpha^{'}Y_1, \alpha^{'}Y_2, \ldots, \alpha^{'}Y_n.$$
Now, this means that $\alpha^{'}X_1, \alpha^{'}X_2, \ldots, \alpha^{'}X_m \sim N(\alpha^{'}\mu_1, \alpha^{'}\Sigma\alpha)$.
Similarly, $\alpha^{'}Y_1, \alpha^{'}Y_2, \ldots, \alpha^{'}Y_n \sim N(\alpha^{'}\mu_2, \alpha^{'}\Sigma\alpha)$.
Now, my null hypothesis is $H_0 : \alpha^{'}\mu_1 = \alpha^{'}\mu_2$ versus the alternative hypothesis $H_1 : \alpha^{'}\mu_1 > \alpha^{'}\mu_2$.
My question is : how do I prove that the power function of the one-sided Mann Whitney U Test will be maximum when $\alpha \propto \Sigma^{-\frac{1}{2}} (\mu_1 - \mu_2)$ ?