Self-dual representations of SL(2,F): an approach using the Iwahori–Hecke algebra

Abstract

Let F be a non-Archimedean local field and $G=\text{SL}(2,F).$ Let $(\pi ,V)$ be an irreducible smooth Iwahori-spherical representation of G. It is easy to see that such representations are always self-dual. The space V of π admits a non-degenerate G-invariant bilinear form $( , )$ which is unique up to scaling. It can be shown that the form $( , )$ is either symmetric or skew-symmetric and we set $\epsilon (\pi )=\pm 1$ accordingly. In this article, we use the Bernstein–Lusztig presentation of the Iwahori–Hecke algebra of G and show that $\epsilon (\pi )=1.$

Keywords:

• Iwahori subgroups
• self-dual representations
• signs

2010 Mathematics Subject Classification:

• 20G05
• 22E50

Additional information

Funding

Research of Kumar Balasubramanian was supported by DST-SERB Grant: YSS/2014/000806.