On the Lebesgue Component of Semiclassical Measures for Abelian Quantum Actions

For a large class of symplectic integer matrices, the action on the torus extends to a symplectic ${\mathbb{Z}}^r$-action with $r\ge 2$. We apply this to the study of semiclassical measures for joint eigenfunctions of the quantization of the symplectic matrices of the ${\mathbb{Z}}^r$-action. In the irreducible setting, we prove that the resulting probability measures are convex combinations of the Lebesgue measure with weight $\ge 1\mathrm{/}2$ and a zero entropy measure. We also provide a general theorem in the reducible case showing that the Lebesgue components along isotropic and symplectic invariant subtori must have total weight $\ge 1\mathrm{/}2$.