  Consider a standard representation $\sigma$ of a quasi-split reductive p-adic
group G. The generalized injectivity conjecture, posed by Casselman--Shahidi,
asserts that any generic irreducible subquotient $\pi$ of $\sigma$ is
necessarily a subrepresentation of $\sigma$. We will prove this conjecture,
improving on the verification for many groups by Dijols.
  We first replace it by a more general ``standard submodule conjecture", where
G does not have to be quasi-split and the genericity of $\pi$ is replaced by
the condition that the Langlands parameter of $\pi$ is open.
  We study this standard submodule conjecture via reduction to Hecke algebras.
It does not suffice to pass from G to an affine Hecke algebra, we further
reduce to graded Hecke algebras and from there to algebras defined in terms of
certain equivariant perverse sheaves. To achieve all these reduction steps one
needs mild conditions on the parameters of the involved Hecke algebras, which
have been verified for the majority of reductive p-adic groups and are expected
to hold in general.
  It is in the geometric setting of graded Hecke algebras from local systems on
nilpotent orbits that we can finally put the ``open" condition on L-parameters
to good use. The closure relations between the involved nilpotent orbits
provide useful insights in the internal structure of standard modules, which
highlight the representations associated with open L-parameters. In the same
vein we show that, in the parametrization of irreducible modules of geometric
graded Hecke algebras, generic modules always have ``open L-parameters".
  This leads to a proof of our standard submodule conjecture for graded Hecke
algebras of geometric type, which is then transferred to reductive p-adic
groups. As a bonus, we obtain that the generalized injectivity conjecture also
holds with ``tempered" or ``essentially square-integrable" instead of generic.
