Query

[Johannes Huebschmann <huebschm@math.univ-lille1.fr>]

Dear All

Given a Lie group G and a G-representation V, according
to Hochschild-Mostow, the
ordinary Eilenberg-Mac Lane construction, suitably interpreted
in terms of smooth functions, yields a differentiably injective
resolution of V over G. More precisely, the Eilenberg-Mac Lane
construction (dual bar construction) arises here as the differentiable
cosimplicial G-module having, in degree p,
the space of smooth V-valued maps on a product of p+1 copies
of G, with the ordinary coface and codegeneracy operators.

Suppose now that G is connected and finite-dimensional and let
K be a maximal compact subgroup. Hochschild-Mostow have also shown that the
V-valued differential forms on G/K then yield an injective
resolution of V over G as well. This kind of construction actually
goes back to van Est.

The standard procedure yields comparison maps between the two resolutions.
In degree zero the comparison is, of course, achieved by the obvious map
from C^{\infty}(G/K,V) to C^{\infty}(G,V) induced by the projection
from G to G/K and by the obvious map
from C^{\infty}(G,V) to C^{\infty}(G/K,V) induced by integration over K.

Does anybody know whether, in the literature, the constituents of
a comparison map in higher degrees have been spelled out explicitly
somewhere?

Many thanks in advance

Best regards

Johannes



HUEBSCHMANN Johannes
Professeur de Mathematiques
USTL, UFR de Mathematiques
UMR 8524 Laboratoire Paul Painleve
F-59 655 Villeneuve d'Ascq Cedex  France
http://math.univ-lille1.fr/~huebschm

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