On a twisted Euler-Poincar\'e pairing for graded affine Hecke algebras

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.