Maximal Unramified Tori in Symplectic Groups

TL;DR

This paper compares two different parameterizations of maximal unramified tori in symplectic groups, connecting Bruhat-Tits theory with partition-based descriptions for a deeper understanding of their structure.

Contribution

It establishes a comparison between DeBacker's and Waldspurger's parameterizations specifically for symplectic groups, clarifying their relationship.

Findings

Demonstrates correspondence between the two parameterizations.

Provides explicit descriptions for symplectic groups.

Enhances understanding of unramified tori structure.

Abstract

For a reductive group over a $p$-adic field, DeBacker gives a paramaterization of the conjugacy classes of maximal unramified tori using Bruhat-Tits theory. On the other hand, for unramified classical groups, Waldspurger gives a parameterization in terms of triples of partitions by constructing a regular semisimple element whose structure is governed by the parts of the partitions. In this paper, we compare these two parameterizations in the case of the symplectic group Sp$_{2n}$.