Selfdual cuspidal representations of ${\rm GL}(N)$ and distinction by an inner involution

Abstract: Let $n$ be a positive integer, $F$ be a non-Archimedean locally compact field of odd residue characteristic $p$ and $G$ be an inner form of GL$(2n,F)$. This is a group of the form GL$(r,D)$ for a positive integer $r$ and division $F$-algebra $D$ of reduced degree $d$ such that $rd=2n$. Let $K$ be a quadratic extension of $F$ in the algebra of matrices of size $r$ with coefficients in $D$, and $H$ be its centralizer in $G$. We study selfdual cuspidal representations of $G$ and their distinction by $H$, that is, the existence of a nonzero $H$-invariant linear form on such representations. When $F$ has characteristic 0, we characterize distinction by $H$ for cuspidal representations of $G$ in terms of their Langlands parameter, proving in this case a conjecture by Prasad and Takloo-Bighash.

number theoryrepresentation theory

Audience: researchers in the topic

Number Theory and Representations in Valparaiso