  For $E/F$ quadratic extension of local fields and $G$ a reductive algebraic
group over $F$, the paper formulates a conjecture classifying irreducible
admissible representations of $G(E)$ which carry a $G(F)$ invariant linear
form, and the dimension of the space of these invariant forms, in terms of the
Langlands parameter of the representation. The paper studies parameter spaces
of Langlands parameters, and morphisms between them associated to morphisms of
$L$-groups. The conjectural answer to the question on the space of
$G(F)$-invariant linear forms is in terms of fibers of a particular finite map
between parameter spaces.
