An explicit matching theorem for level zero discrete series of unit groups of p-adic simple algebras, preprint

Abstract. For A|F a central simple algebra over a p-adic local field the group of units A × ∼ = GLm(Dd) is a general linear group over a central division algebra Dd|F of index d. The product n = dm being fixed, the Abstract Matching Theorem (AMT) implies the existence of bijective maps JL between the sets of discrete series representations of the groups A × such that a character relation is preserved. In this paper we construct maximal level zero extended type components for every level zero discrete series representation of A×. Its maximal level zero extended type determines the discrete series representation uniquely (without any twist ambiguities as for the usual types) and, conversely, every level zero discrete series representation Π contains a maximal level zero extended type component Σ̃(Π) which is unique up to conjugacy. In order to determine how JL matches the extended types we find certain regular elliptic elements where the characters of Σ̃(Π) and Π are the same and we compute the character values at these elements by using a version of Shintani descent which we develop in Appendix B. Surprisingly, we find that AMT also implies explicit Shintani descent for irreducible characters of finite general linear groups which have cuspida