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 $r\ge 2$, we prove $\dim C_r(\mathcal{N}(\mathfrak{p}))\le \lfloor r\dim \mathfrak{p}\mathrm{/}2\rfloor$ and show that every irreducible component is isotropic for a natural family of alternating forms. For a nonzero Hermitian pair with $\mathfrak{g}$ simple and $\mathfrak{p}={\mathfrak{p}}^{-}\oplus {\mathfrak{p}}^{+}$, put $d=\dim {\mathfrak{p}}^{+}$ and $\mathrm{\ell }=\mathrm{rk}(\mathfrak{g},\mathfrak{k})$. Then $$\dim C_r(\mathcal{N}(\mathfrak{p}))=rd,\dim C_r(\mathfrak{p})=\max \{rd,2d+(r-1)\mathrm{\ell }\}\mathrm{.}$$ In this case we also determine the maximal-dimensional components of $C_r(\mathfrak{p})$; for $r\ge 3$, the only maximal-dimensional nilpotent components are $({\mathfrak{p}}^{+}{)}^r$ and $({\mathfrak{p}}^{-}{)}^r$. For arbitrary symmetric pairs, $\dim C_r(\mathcal{N}(\mathfrak{p}))=ra+m$ for all large $r$, where $a$ is the largest dimension of an abelian subspace of $\mathfrak{p}$ consisting of nilpotent elements and $m$ is the dimension of the variety of such subspaces. Applications give dimension and component formulas for equivariant punctual Quot schemes under explicit framing conditions, and bounds for punctual Gorenstein loci refined by socle parity.