The Morava E-theories of finite general linear groups

TL;DR

This paper explores the rational Morava E-theory of classifying spaces of finite general linear groups using representation theory, formal group laws, and generalized characters, providing new insights into their structure.

Contribution

It introduces novel methods to analyze Morava E-theories of finite groups by combining representation theory and formal group law techniques, extending previous work.

Findings

Detailed analysis of Morava E-theory for groups of dimension ≤ p

New integral results based on formal group law theory

Enhanced understanding of classifying spaces of finite general linear groups

Abstract

By studying the representation theory of a certain infinite $p$-group and using the generalised characters of Hopkins, Kuhn and Ravenel we find useful ways of understanding the rational Morava $E$-theory of the classifying spaces of general linear groups over finite fields. Making use of the well understood theory of formal group laws we establish more subtle results integrally, building on relevant work of Tanabe. In particular, we study in detail the cases where the group has dimension less than or equal to the prime $p$ at which the $E$-theory is localised.