ÄQUIVARIANTE η-INVARIANTEN HOMOGENER RÄUME

English Summary

In our Ph-D thesis, we calculate the G-equivariant η-invariant of a family of generalized Dirac operators on a quotient M = G/H of compact connected Lie groups. More precisely: If M = G/H is G-spin, let ΣM be the G-equivariant spinor bundle of M, let U^{κ}M be the homogeneous vector bundle induced by the irreducible unitary H-representation with heighest weight κ and let D_{M}^{κ} be the natural Dirac operator on ΣM ⊗ U^{κ}M. We construct a “reductive” Dirac operator D̃_{M}^{κ} which agrees with D_{M}^{κ} up to zero-order terms.

Now let T ⊃ S be maximal tori of G and H and suppose that G/T is G-spin as well. Then one can reduce the computation of η_{G}(D̃_{M}^{κ}) to the torus T/S:

η_{G}(D̃_{M}^{κ})(g, s) = η_{G}(D̃_{G/S}^{κ+ρ_{H}} )(g, s) = \frac{A_{G}(η_{T} (D̃_{T/S}^{κ+ρ_{H}})( · , s))}{A_{G}(ρ_{G})} (g)

for g ∈ T and s ∈ ℂ; here ρ_{H} and ρ_{G} are half sums of positive roots, and A_{G} denotes the alternating sum over the Weyl group of G. By computing η_{T} (D̃_{T/S}^{κ+ρ_{H}}) one gets an explicit formula for η_{G}(D̃_{M}^{κ}) which holds with slight modifications if G/T or M are not G-spin. Furthermore, η_{G}(D̃_{M}^{κ}) vanishes identically if rk G − rk H is larger than 1.

The function η_{G}(D̃_{M}^{κ}) “approximates” η_{G}(D_{M}^{κ}) (in a non-technical sense): Restricted to the subset G_{0} of G that acts freely on M, η_{G}(D̃_{M}^{κ}) − η_{G}(D_{M}^{κ}) is a virtual character of G, which vanishes if D_{M} is the untwisted Dirac operator on M for a normal metric. Thus the formula from above can be applied to compute some ℝ/ℤ-valued invariants of flat bundles over locally homogeneous spaces of the type Γ\G/H.

If M is a compact symmetric space we have D̃_{M}^{κ} = D_{M}^{κ}. We calculate the equivariant η-invariants of the untwisted Dirac operator and of the tangential boundary operator of the signature complex for all irreducible symmetric spaces with rk G − rk H = 1 (i.e., for the odd dimensional real Grassmannians and for SU_{3}/SO_{3}).

As a function on G, the equivariant η-function η_{G}(D) can have singularities at all elements g ∈ G that don’t act freely. If M = ∂N for some manifold N and if D and the G-action can be extended to N in a compatible way, then these singularities can be described by an integral over a certain equivariant differential form on the fixed point set of g on M. One can thus recover η_{G}(D_{M}) on all of G if one only knows its values on G_{0}. We use this method together with our formula for η_{G}(D̃_{M}) to calculate η_{G}(D_{M}) for the Aloff-Wallach spaces M = SU_{3}/S^{1}.

Sebastian Goette, Math. Inst. der Univ. Freiburg, Eckerstr. 1, 79104 Freiburg, Germany

   E-mail address: goette@mathematik.uni-freiburg.de

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