  We study a twisted Euler-Poincar\'e pairing for graded affine Hecke algebras,
and give a precise connection to the twisted elliptic pairing of Weyl groups
defined by Ciubotaru-He. The Ext-groups for an interesting class of
parabolically induced modules are also studied in a connection with the twisted
Euler-Poincar\'e pairing. We also study a certain space of graded Hecke algebra
modules which equips with the twisted Euler-Poincar\'e pairing as an inner
product.
