  We consider $\beta$--Plancherel measures \cite{Ba.Ra.} on subsets of
partitions -- and their asymptotics. These subsets are the Young diagrams
contained in a $(k,\ell)$--hook, and we calculate the asymptotics of the
expected shape of these diagrams, relative to such measures. We also calculate
the asymptotics of the distribution function of the lengths of the rows and the
columns for these diagrams. This might be considered as the restriction to the
$(k,\ell)$--hook of the fundamental work of Baik, Deift and Johansson
\cite{B.D.J.1}. The above asymptotics are given here by ratios of certain
Selberg-type multi--integrals.
