Representations of Quivers of TypeAand the Multisegment Duality

✍ Scribed by Harold Knight; Andrei Zelevinsky

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 613 KB
• Volume: 117
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1996.0013

✦ Synopsis

In this paper we study a certain involution : Z S + Ä Z S + which we call the multisegment duality. This involution acts on the free abelian semigroup Z S + whose generators are indexed by the set S=S r of pairs of integers (i, j) such that 1 i j r (throughout the paper, r will be a fixed positive integer). The involution first appeared in [11], [12] in the context of the Deligne Langlands correspondence for the groups GL n over a p-adic field. Roughly speaking, the elements of Z S + serve as labels for the irreducible representations of the spherical principal series (or, equivalently, for the irreducible representations of the affine Hecke algebra); the involution describes the well-known duality interchanging the identity and the Steinberg representations. Later was studied in [8] (under the name ``Zelevinsky involution''); recently in [1] it was extended to other reductive p-adic groups.

The involution has another important interpretation in terms of the canonical bases for quantum groups. Under this interpretation, the elements of Z S + serve as labels for the canonical basis vectors in U + , the q-deformation of the universal enveloping algebra of the Lie algebra of nilpotent upper triangular (r+1)\_(r+1) matrices (see, e.g., [6], [3]). Now describes the action on the canonical basis by a natural antiautomorphism of U + (more details are given in Section 4 below).