The distribution of the latent roots of the covariance matrix calculated from a sample from a normal multivariate population, was found by Fisher , Hsu  and Roy  for the special, but important case when the population covariance matrix is a scalar matrix, $\Sigma = \sigma^2I$. By use of the representation theory of the linear group, we are able to obtain the general distribution for arbitrary $\Sigma$.
"The Distribution of the Latent Roots of the Covariance Matrix." Ann. Math. Statist. 31 (1) 151 - 158, March, 1960. https://doi.org/10.1214/aoms/1177705994