  By the theory of Colmez and Fontaine, a de Rham representation of the Galois
group of a local field roughly corresponds to a representation of the
Weil-Deligne group equipped with an admissible filtration on the underlying
vector space. Using a modification of the classical local Langlands
correspondence, we associate with any pair consisting of a Weil-Deligne group
representation and a type of a filtration (admissible or not) a specific
locally algebraic representation of a general linear group. We advertise the
conjecture that this pair comes from a de Rham representation if and only if
the corresponding locally algebraic representation carries an invariant norm.
In the crystalline case, the Weil-Deligne group representation is unramified
and the associated locally algebraic representation can be studied using the
classical Satake isomorphism. By extending the latter to a specific norm
completion of the Hecke algebra, we show that the existence of an invariant
norm implies that our pair, indeed, comes from a crystalline representation. We
also show, by using the formalism of Tannakian categories, that this latter
fact is compatible with classical unramified Langlands functoriality and
therefore generalizes to arbitrary split reductive groups.
