Langlands parameters associated to special maximal parahoric spherical representations

We describe the image, under the local Langlands correspondence for tori, of the characters of a torus which are trivial on its Iwahori subgroup. Let $k$ be a non-archimedian local field. Let $\boldsymbol{G}$ be a connected reductive group defined over $k$, which is quasi-split and split over a tamely ramified extension. Let $K$ be a special maximal parahoric subgroup of $\boldsymbol{G}(k)$. To the class of representations of $\boldsymbol{G}(k)$, having a non-zero vector fixed under $K$, we establish a bijection, in a natural way, with the twisted semi-simple conjugacy classes of the inertia fixed subgroup of the dual group $\hat{\boldsymbol{G}}$. These results generalize the well known classical results to the ramified case.