On the module structure of the center of hyperelliptic Krichever-Novikov algebras II

Let $R := R_{2}(p)=$\mathbb{C}$[t^{\pm 1}, u : u^2 = t(t-\alpha_1)\cdots (t-\alpha_{2n})] $ be the coordinate ring of a nonsingular hyperelliptic curve and let $\mathfrak{g}\otimes R$ be the corresponding current Lie algebra. \color{black} Here $\mathfrak{g}$ is a finite dimensional simple Lie algebra defined over $\mathbb{C}$ and \begin{equation*} p(t)= t(t-\alpha_1)\cdots (t-\alpha_{2n})=\sum_{k=1}^{2n+1}a_kt^k. \end{equation*} In earlier work, Cox and Im gave a generator and relations description of the universal central extension of $\mathfrak{g}\otimes R$ in terms of certain families of polynomials $P_{k,i}$ and $Q_{k,i}$ and they described how the center ${\mathrm{\Omega }}_R\mathrm{/}dR$ of this universal central extension decomposes into a direct sum of irreducible representations when the automorphism group was the cyclic group $C_{2k}$ or the dihedral group $D_{2k}$. We give examples of $2n$-tuples $({\alpha }_1,\dots ,{\alpha }_{2n})$, which are the automorphism groups $\mathbb G_n=$\text{Dic}${n}$,$\mathbb U_n\cong D_n$($n$odd),or$\mathbb U_n$($n$ even) of the hyperelliptic curves \begin{equation} S=$\mathbb{C}$[t, u: u^2 = t(t-\alpha_1)\cdots (t-\alpha{2n})] \end{equation} given in [CGLZ17]. In the work below, we describe this decomposition when the automorphism group is ${\mathbb{U}}_n=D_n$, where $n$ is odd.