Past exam of the chemistry A course of the University of Cambridge / 2022 / Paper 1 / A5 / c / ii / Solution

The overall vibrational ground state is invariant under every molecular symmetry operation, so it transforms as $A_{1g}$. Combining it with the electronic ground state givesFor a degenerate harmonic normal mode, the zero-quantum wavefunction depends on the invariant sum $Q_x^2+Q_y^2$, so the ground vibrational symmetry remains totally symmetric even when excited vibrational states form multiplets.