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

TL;DR

This paper introduces a twisted Euler-Poincaré pairing for graded affine Hecke algebras, linking it to twisted elliptic pairings of Weyl groups, and explores Ext-groups and module spaces related to this pairing.

Contribution

It establishes a precise connection between the twisted Euler-Poincaré pairing and the twisted elliptic pairing of Weyl groups, and analyzes Ext-groups for parabolically induced modules.

Findings

Connection between twisted Euler-Poincaré and elliptic pairings established

Ext-groups for certain modules studied in relation to the pairing

A module space with the pairing as an inner product analyzed

Abstract

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.