On the duality involution for p-adic general spin groups

Abstract

Let GSpinm be the quasi-split general spin group over a p-adic field F for a positive integer m. For a certain class of quasi-split reductive groups, including the general spin groups, a specific involution ι, called the duality involution, is constructed in [Pra19]–generalising the MVW involution on the classical groups. In this article, for any irreducible admissible representation (π,V) of GSpinm(F), we show that π∨≃πι. Here π∨ is the contragredient of π and πι is the composite of π with ι. We also prove the analogue of this result for the split general spin groups over Fq((t)).