  In this paper, we establish connections between the first extensions of
simple modules and certain filtrations of of standard modules in the setting of
graded Hecke algebras. The filtrations involved are radical filtrations and
Jantzen filtrations. Our approach involves the use of information from the
Langlands classification as well as some deeper understanding on some structure
of some modules. Such module arises from the image of a Knapp-Stein type
intertwining operator and is a quotient of a generalized standard module. Along
the way, we also deduce some results on the blocks for finite-dimensional
modules of graded Hecke alebras.
  As an application, we compute the Ext-groups for irreducible modules in a
block for the graded Hecke algebra of type $C_3$, assuming the truth of a
version of Jantzen conjecture.

Representation Theory