  As a consequence of his numerical local Langlands correspondence for $GL(n)$,
Henniart deduced the following theorem: If $F$ is a nonarchimedean local field
and if $\pi$ is an irreducible admissible representation of $GL(n,F)$, then,
after a finite sequence of cyclic base changes, the image of $\pi$ contains a
vector fixed under an Iwahori subgroup. This result was indispensable in all
proofs of the local Langlands correspondence. Scholze later gave a different
proof, based on the analysis of nearby cycles in the cohomology of the
Lubin-Tate tower.
  Let $G$ be a reductive group over $F$. Assuming a theory of stable cyclic
base change exists for $G$, we define an incorrigible supercuspidal
representation $\pi$ of $G(F)$ to be one with the property that, after any
sequence of cyclic base changes, the image of $\pi$ contains a supercuspidal
member. If F is of positive characteristic then we define $\pi$ to be pure if
the Langlands parameter attached to $\pi$ by Genestier and Lafforgue is pure in
an appropriate sense. We conjecture that no pure supercuspidal representation
is incorrigible. We prove this conjecture for $GL(n)$ and for classical groups,
using properties of standard $L$-functions; and we show how this gives rise to
a proof of Henniart's theorem and the local Langlands correspondence for
$GL(n)$ based on V. Lafforgue's Langlands parametrization, and thus independent
of point-counting on Shimura or Drinfel'd modular varieties.
