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 $Omega_R/dR$ 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 $mathbb G_n=text{Dic}_{n}$, $mathbb U_ncong D_n$ ($n$ 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.