Global Iwasawa-decomposition of SL($n$, $\mathbb{A}_{\mathbb{Q}}$)

TL;DR

This paper provides a comprehensive analysis of the Iwasawa-decomposition for matrices in SL(n, Q_p) and SL(n, R), including algorithms, formulas, and proofs of uniqueness and parameterization.

Contribution

It introduces an algorithm for computing the Iwasawa-decomposition in SL(n, Q_p), proves the uniqueness of p-adic norms across decompositions, and offers a global formula relating these norms to Plücker coordinates.

Findings

Algorithm for Iwasawa-decomposition in SL(n, Q_p)

Proof of p-adic norm uniqueness across decompositions

Global formula relating Cartan torus norms to Plücker coordinates

Abstract

We discuss the Iwasawa-decomposition of a general matrix in SL($n$, $\mathbb{Q}_p$) and SL($n$, $\mathbb{R}$). For SL($n$, $\mathbb{Q}_p$) we define an algorithm for computing a complete Iwasawa-decomposition and give a formula parameterizing the full family of decompositions. Furthermore, we prove that the $p$-adic norms of the coordinates on the Cartan torus are unique across all decompositions and give a closed formula for them which is proven using induction. For the case SL($n$, $\mathbb{R}$), the decomposition is unique and we give formulae for the complete decomposition which are also proven inductively. Lastly we outline a method for deriving the norms of the coordinates on the Cartan torus in the framework of representation theory. This yields a simple formula valid globally which expresses these norms in terms of the vector norms of generalized Pl\"ucker coordinates.