  Consider the irreducible representations of a real reductive group
$G(\mathbb{R})$, and their parametrization by the local Langlands
correspondence. We ask: does the parametrization give easily accessible
information on the restriction of representations to a maximal compact subgroup
$K(\mathbb{R})$ of $G(\mathbb{R})$? We find a natural connection between the
set of lowest $K$-types of a representation and its Langlands parameters.
  For our results, it is crucial to use the refined version of the local
Langlands correspondence, involving (coverings of) component groups attached to
$L$-homomorphisms. The first part of the paper is a simplified description of
this refined parametrization.
