Dimensions and maximal-dimensional components of higher commuting varieties of symmetric pairs

Let $\mathfrak p$ be the $(-1)$-eigenspace of a complex semisimple symmetric pair. For every $r\ge2$, we prove $\dim C_r(\mathcal N(\mathfrak p))\le \lfloor r\dim\mathfrak p/2\rfloor$ and show that every irreducible component is isotropic for a natural family of alternating forms. For a nonzero Herm