The Struture of an Unramified $L$-Packet and Related Results

Let $\mathrm G$ be a connected reductive group over
a nonarchimedean local field $F$, such as $\mathbb{Q}_p$. The local
Langlands conjectures predict that the (isomorphism classes
of) irreducible `admissible'
representations of $\mathrm G(F)$ (in vector spaces over
$\mathbb{C}$) can be partitioned in a certain natural way
into equivalence classes, called $L$-packets.

Here we will consider an `unramified'
$\mathrm G$, and the unramified representations of
$\mathrm G(k)$. Such representations can be parameterized
by pairs $(K, \lambda)$, where $K$ is a so called
`hyperspecial' compact subgroup of $\mathrm G(k)$ and
$\lambda$ is a character of a maximally split maximal torus
$\mathrm T$ of $\mathrm G$. Let $\tau_{K, \lambda}$
be the representation associated to a pair $(K, \lambda)$.

We will first describe when two such representations
$\tau_{K, \lambda}$ and $\tau_{K', \lambda'}$ are isomorphic.
Now given an unramified $L$-packet we can :
(i) think of its elements as
$\tau_{K, \lambda}$ for certain (equivalence classes of)
pairs $(K, \lambda)$; and
(ii) realize the set of these elements as a principal
homogeneous space for a certain natural finite abelian group.

We will describe how the parametrization in (ii) is related
to the description in (i).


It has been well known for a while that, using the so called
Satake isomorphism or otherwise, one can attach to an unramified representation of an unramified group $\mathrm G$
a Frobenius-twisted semisimple conjugacy class in the Langlands dual
group $\hat G$ of $\mathrm G$. We will generalize this result to
an analogous class of representations of a {\em tamely ramified}
group $\mathrm G$. The key step in the proof is a description of the
image under the local Langlands correspondence of the set of
characters of a torus that are trivial on its `Iwahori subgroup'.