$e$-invariants of quotients of Lie groups

Abstract

Let $G$ be a simply connected compact Lie group and $\mathcal{L}$ be the left invarinat framing of $G$. Let ${\mathcal{L}}^{\lambda }$ be the framing obtained by twisting $\mathcal{L}$ by a faithful representation $\lambda$. Given a torus subgroup $T^{\prime \prime }$ of $G$ we have a framing $({\mathcal{L}}^{\lambda }{)}_{T^{\prime \prime }}$ of the quotient $G/T^{\prime \prime }$ induced from ${\mathcal{L}}^{\lambda }$. In this note we show that under a certain dimensional condition the $e_{\mathbb{C}}$-invariant of $G/T^{\prime \prime }$ 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.

adds one remark (Remark 2) in order to improve the explanation of the framing of $G/T"$ on page 1