Representations of rank two affine Hecke algebras at roots of unity.
Let \(\mathcal H:=\mathcal H(q)\) denote an affine Hecke algebra with parameter \(q\) associated to a given root system. \(\mathcal H\) contains a polynomial subalgebra \(\mathbb C[X]\) with generators corresponding to weights for the associated root system. Each character \(t\colon\mathbb C[X]\to\mathbb C\) determines a one-dimensional \(\mathbb C[X]\)-module and, by induction, an \(\mathcal H\)-module denoted \(M(t)\). Such a module \(M(t)\) is known as a principal series module. It is known that the irreducible \(\mathcal H\)-modules arise as quotients of principal series modules. The goal of this work is to give a complete description of the irreducible \(\mathcal H\)-modules for root systems of types \(A_2\), \(C_2\), and \(G_2\). New information is obtained in particular in the latter two cases when \(q\) is a root of unity. The description is obtained by giving a combinatorial geometric description of the composition factors of the principal indecomposable modules.NEWLINENEWLINE This paper builds on the foundational work of \textit{A. Ram} [in: Advances in algebra and geometry. Proceedings of the international conference on algebra and geometry, Hyderabad, India, 2001. New Delhi: Hindustan Book Agency. 57-91 (2003; Zbl 1062.20006)]. For a character \(t\), the author gives a geometric picture of the weight spaces of \(M(t)\) using the geometry of the Weyl chambers for the underlying root system. In particular, one gets a visual illustration of when weight spaces are part of the same composition factor. A key role is played by the value of \(t(X^\alpha)\) where \(X^\alpha\in\mathcal H\) is the polynomial generator corresponding to a positive root \(\alpha\). Much of the structure is determined by two sets of positive roots: \(\{\alpha\mid t(X^\alpha)=1\}\) and \(\{\alpha\mid t(X^\alpha)=q^{\pm 2}\}\).
scientific article; zbMATH DE number 6055212