We study ineffability, the Shelah property, and indescribability of $\mathscr{P}_\kappa \lambda$ when ${\rm cf}(\lambda)\kappa$ . We prove that if $\lambda$ is a strong limit cardinal with ${\rm cf}(\lambda)\kappa$ then the ineffable ideal, the Shelah ideal, and the completely ineffable ideal over $\mathscr{P}_\kappa \lambda$ are the same, and that it can be precipitous. Furthermore we show that $\Pi^1_1$ -indescribability of $\mathscr{P}_\kappa \lambda$ is much stronger than ineffability if $2^\lambda=\lambda^{\kappa}$ .