On Simple $\mathbb{Z}_2$-Invariant and Corner Function Germs

V. I. Arnold has classified simple (i.e., having no moduli for the classification) singularities (function germs), and also simple boundary singularities: function germs invariant with respect to the action $\sigma(x_1;y_1,\dots,y_n)=(-x_1;y_1,\dots,y_n)$ of the group $\mathbb{Z}_2$. In particular, it was shown that a function germ (a boundary singularity germ) is simple if and only if the intersection form (respectively, the restriction of the intersection form to the subspace of anti-invariant cycles) of a germ in $3+4s$ variables stable equivalent to the one under consideration is negative definite and if and only if the (equivariant) monodromy group on the corresponding subspace is finite. We formulate and prove analogs of these statements for function germs invariant with respect to an arbitrary action of the group $\mathbb{Z}_2$, and also for corner singularities.