# Equivariant APS index for Dirac operators of non-product type near the boundary

## TL;DR

This paper develops a generalized equivariant index theorem for Dirac operators that are not of product type near the boundary, providing explicit formulas and applications to specific geometric cases like SKR manifolds.

## Contribution

It introduces a delocalized equivariant index theorem for non-product boundary Dirac operators and computes explicit formulas for special geometric cases.

## Key findings

- Derived a Kirillov formula for non-product boundary Dirac operators.
- Provided explicit formulas for equivariant signatures and eta-invariants.
- Established vanishing results for certain SKR manifolds.

## Abstract

We consider a generalized APS boundary problem for a G-invariant Dirac-type operator, which is not of product type near the boundary. We establish a delocalized version (a so-called Kirillov formula) of the equivariant index theorem for this operator. We obtain more explicit formulas for different geometric Dirac-type operators. In particular, we get a formula for the equivariant signature of a local system over a manifold with boundary. In case of a trivial local system, our formula can be viewed as a new way to compute the infinitesimal equivariant eta-invariant of S. Goette. We explicitly compute all the terms in this formula, which involve the equivariant Hirzebruch L-form and its transgression, for four-dimensional SKR manifolds, a class including many Kaehler conformally Einstein manifolds, in the case where the boundary is given as the zero level set of a certain Killing potential. In the case of SKR metrics which are local Kaehler products, these terms are zero, and we obtain a vanishing result for the infinitesimal equivariant eta invariant.