$e$-invariants of quotients of Lie groups

TL;DR

This paper investigates the $e$-invariants of quotients of simply connected compact Lie groups, showing they generate the $J$-homomorphism or twice, unifying previous results for specific groups.

Contribution

It establishes a general condition under which the $e_ ext{complex}$-invariant of certain quotients generates the $J$-homomorphism, unifying earlier specific cases.

Findings

The $e_ ext{complex}$-invariant can generate the $J$-homomorphism under certain conditions.

Provides a unified proof for known results on $SU(2n)$, $Spin(4n+1)$, and $Spin(8n-2)$.

Connects $e$-invariants with the structure of Lie group quotients.

Abstract

Let $G$ be a simply connected compact Lie group and $\mathscr{L}$ be the left invarinat framing of $G$. Let $\mathcal{L}^\lambda$ be the framing obtained by twisting $\mathscr{L}$ by a faithful representation $\lambda$. Given a torus subgroup $T''$ of $G$ we have a framing $(\mathcal{L}^\lambda)_{T''}$ of the quotient $G/T''$ induced from $\mathcal{L}^\lambda$. In this note we show that under a certain dimensional condition the $e_\mathbb{C}$-invariant of $G/T''$ with this framing provides a generator of the $J$-homomorphism or twice that. Thereby we also give a unified proof of the results for $SU(2n)$, $Spin(4n+1)$ and $Spin(8n-2)$ $(n\ge 1)$ previously proved.