Hecke operators in Morava $E$-theories of different heights

TL;DR

This paper constructs and studies Hecke operators acting on Morava $E$-theories of different heights, revealing how higher height Hecke algebra actions restrict to lower heights and enriching the algebraic structure of these cohomology theories.

Contribution

It introduces Hecke operators in an amalgamated cohomology theory connecting Morava $E$-theories of consecutive heights, extending the known actions and analyzing their algebraic relationships.

Findings

Constructed Hecke operators in an amalgamated cohomology theory.

Established the action of $ ext{Hecke}_{n+1}$ on the $n$th Morava $E$-theory.

Showed the $ ext{Hecke}_{n+1}$-module structure is a restriction of the $ ext{Hecke}_n$-module structure.

Abstract

There is a natural action of a kind of Hecke algebra $\mathcal{H}_n$ on the $n$th Morava $E$-theory of spaces. We construct Hecke operators in an amalgamated cohomology theory of the $n$th and the $(n+1)$st Morava $E$-theories. These operations are natural extensions of the Hecke operators in the $(n+1)$st Morava $E$-theory, and they induce an action of the Hecke algebra $\mathcal{H}_{n+1}$ on the $n$th Morava $E$-theory of spaces. We study a relationship between the actions of the Hecke algebras $\mathcal{H}_n$ and $\mathcal{H}_{n+1}$ on the $n$th Morava $E$-theory, and show that the $\mathcal{H}_{n+1}$-module structure is obtained from the $\mathcal{H}_n$-module structure by the restriction along an algebra homomorphism from $\mathcal{H}_{n+1}$ to $\mathcal{H}_n$.