Kernel and cokernel characterization of duality operators
Determine the kernel and cokernel of the strongly symmetric duality operator \mathcal{D}^{(g)} between H^g and \tilde{H}: prove that \mathrm{ker}\,\mathcal{D}^{(g)} = \mathscr{H} / \mathscr{H}_g^{\hat{1}} and that \mathrm{coker}\,\mathcal{D}^{(g)} = \tilde{\mathscr{H}} / \tilde{\mathscr{H}}^{c} for all g in the conjugacy class c.
References
From eq:DdaggerD and eq:DDdagger, we realize that both $(\mathcal{D}{(g)} )\dag \mathcal{D}{(g)}$ and $\mathcal{D}{(g)} (\mathcal{D}{(g)} )\dag$ are proportional to a projector in Hilbert spaces $\mathscr{H}_g$ and $\tilde{\mathscr{H}}$. We therefore conjecture that the kernel of the duality operator (with or without twist) is precisely
\begin{equation}
\mathrm{ker}\, \mathcal{D}{(g)} = \mathscr{H} / \mathscr{H}_g{\hat{1} \;, \quad \forall g \in c \;, \label{eq:kernelD}
\end{equation}
and analogously the cokernel of the duality operator becomes
\begin{equation}
\mathrm{coker}\, \mathcal{D}{(g)} =
\mathrm{ker}\, \left(\mathcal{D}{(g)}\right){\dagger} =\tilde{\mathscr{H} / \tilde{\mathscr{H}{c} \;, \quad \forall g \in c \;.
\label{eq:cokernelD}
\end{equation}