Let $\mathfrak{ g}$ be an affine Lie algebra with index set $I = \{ 0, 1, 2, \cdots , n\}$ and $\mathfrak{ g}^L$ be its Langlands dual. It is conjectured that for each Dynkin node $i \in I \setminus \{ 0\}$ the affine Lie algebra $\mathfrak{ g}$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of a certain coherent family of perfect crystals for the Langland dual $\mathfrak{ g}^L$. In this paper we construct positive geometric crystals for $\mathcal{ V}(C_n^{ (1)})$ in the level zero fundamental spin $C_n^{ (1)}$- module $W(\varpi_n)$ for $n = 2, 3,4$ and show that its ultra-discretization is isomorphic to the limit $B^{ n, \infty}$ of a coherent family $\{ B^{ n, l}\}_{ l \geq 1}$ of perfect crystals for the Langland dual $D_n^{ (2)}$ which proves the conjecture in these cases.
Categories: math.RT