Formal degree of principal series of quasi-split groups

TL;DR

This paper proves the formal degree conjecture for discrete series in principal series of quasi-split groups over non-archimedean fields, using types and the local Langlands correspondence.

Contribution

It constructs types for Bernstein components and verifies the formal degree conjecture via a reduction to unipotent representations.

Findings

Proves the formal degree conjecture for a class of representations.

Constructs types for Bernstein components associated with principal series.

Reduces the problem to unipotent representations of other quasi-split groups.

Abstract

Let $\mathcal{G}$ be a quasi-split connected reductive group over a non-archimedean local field $F.$ In this paper, we prove the formal degree conjecture for discrete series representations contained in a principal series of $\mathcal{G}(F)$. We first construct a type for each Bernstein component attached to a principal series representation of $\mathcal{G}(F).$ We then use these types and the local Langlands correspondence for principal series representations defined in [Sol25] to verify the formal degree conjecture. Our approach follows a similar strategy to [Ric25], reducing the problem to the case of unipotent representations of some other quasi-split group.