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

Allan J. Silberger , Ernst-Wilhelm Zink

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 1439-9679), 55

MSC 2000

22E50 Representations of Lie and linear algebraic groups over local fields

Abstract
For $A|F$ a central simple algebra over a ${\frak p}$-adic local field the group of units $A^\times\cong GL_m(D_d)$ is a general linear group over a central division algebra $D_d|F$ of index $d.$ The product $n=dm$ being fixed, the Abstract Matching Theorem (AMT) implies the existence of bijective maps ${\Cal J}\!\!{\Cal L}$ between the sets of discrete series representations of the groups $A^\times$ such that a character relation is preserved. In this paper we construct maximal level zero extended type components for every level zero discrete serie representation of $A^\times$. Its maximal level zero extended typ determines the discrete series representation uniquely (without any twist ambiguities as for the usual types) and, conversely, every level zero discrete series representation $\Pi$ contains a maximal level zero extended type component $\tilde\Sigma(\Pi)$ which is unique up to conjugacy. In order to determine how ${\Cal J}\!\!{\Cal L}$ matches the extended types we find certain regula elliptic elements where the characters of $\tilde\Sigma(\Pi)$ and $\Pi$ 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 cuspidal descent.