$e$-invariants of quotients of Lie groups

TL;DR

The paper studies the complex $e$-invariant $e_\mathbb{C}$ for quotients $G/T''$ of simply connected compact Lie groups under twisted left-invariant framings, aiming to generate the image of the $J$-homomorphism. Under the dimension condition $\dim(G/T'')=4l-1$, it proves $e_\mathbb{C}([G/T'', (\mathcal{L}^{(r-1)\rho_{\mathbb{R}}})_{T''}]) = (-1)^{l-1} B_l/(2l)$, providing a generator up to sign and unifying previous results for $SU(2n)$, $Spin(4n+1)$, and $Spin(8n-2)$. The method hinges on a tensor product decomposition of the line bundle $E$ associated to a circle bundle, explicit $SU(2)$-block constructions, and a careful degree-one map argument that reduces the computation to a Proposition 2.1-type calculation in LS. Consequently, the Bernoulli-number expression links the $e$-invariant to the $J$-image in a broad class of homogeneous quotients, with a clean proof strategy applicable across the listed families and their twisted framings.

Abstract

Let $G$ be a simply connected compact Lie group and $\mathscr{L}$ be the left invarinat framing of $G$. Let $\mathcal{L}^λ$ be the framing obtained by twisting $\mathscr{L}$ by a faithful representation $λ$. Given a torus subgroup $T''$ of $G$ we have a framing $(\mathcal{L}^λ)_{T''}$ of the quotient $G/T''$ induced from $\mathcal{L}^λ$. 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.

Theorems & Definitions (13)

• Theorem 1
• Remark 1
• Theorem 2
• Remark 2
• Lemma 1
• Lemma 2
• proof
• Lemma 3: cf. LS, §2, Example 3
• proof
• Lemma 4