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Let $G$ be a connected, reductive group over a $p$-adic field $k$. Assume $G$ is quasi-split with Borel subgroup $B = TU$. Consider those irreducible admissible representations $\pi$ of $G(k)$ which are isomorphic to a subrepresentation of the unramified principal series representation $I(\chi)$ for some unramified character $\chi$ of $T(k)$. Equivalently, $\pi$ has an Iwahori fixed vector.

The version of local Langlands correspondence stated by A. Borel in Automorphic L-Functions in the Corvallis proceedings would attach to $\pi$ a conjectural homomorphism $\varphi': W_k' \rightarrow \space ^LM$, where $W_k'$ is the Weil-Deligne group of $k$. Is this correspondence known for irreducible subrepresentations of an unramified principal series?

What if we just consider the unramified character $\chi$ itself? Is it known how to attach a homomorphism $\varphi: W_k \rightarrow \space ^LT$ to $\chi$? When $T$ is split, this is just local class field theory.

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  • $\begingroup$ Pardon if I am just confusing myself. If this is about local Langlands for the group $T$ (as a reductive group by its own), then this is the well-established local Langlands for tori, for which I will recommend J.-K. Yu's Ottawa paper "On the local Langlands correspondence for tori" for a reference. In particular unramified $\chi$ corresponds to unramified $\varphi$. $\endgroup$ – Cheng-Chiang Tsai Jul 12 '18 at 23:39

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