**Kumar Balasubramanian**

IISER Bhopal

November 22, 2012

**Self-dual representations with vectors fixed under an Iwahori subgroup**:
An irreducible smooth self-dual representation of a reductive $p$-adic
group preserves either a non-zero symmetric or a nonzero alternating form on the underlying space and accordingly is said to have sign plus one or minus one. There is strong evidence that this sign is always one in the case of representations with non-trivial Iwahori fixed vectors. We will discuss some of this evidence.