Analysis and homological algebra
Given a real semisimple Lie group we have the category of its Harish Chandra modules and the canonical
globalization functors which produce group representations from Harish-Chandra modules.
Further given a discrete (arithmetic, or S-arithmetic) subgroup we can consider
the cohomology of the subgroup with coefficients
in the globalizations of Harish-Chandra modules.
We are interested in structural results such as finitness, Poincare duality, computation in terms of automorphic forms etc.
Our main technique is to employ invariant
differential operators on symmetric spaces in order
to construct acyclic resolutions of the representations.
If the discrete subgroup is cocompact, then one considers the maximal and minimal globalizations.
While the rank-one case is easy (see
Gamma cohomology and the Selberg zeta function)
the general case is based on deep results of Kashiwara/Schmid
(see
Cohomological properties of the canonical globalizations of Harish-Chandra modules)
In this paper we also consider the smooth/distribution vector globalizations which lead to the same cohomology.
In the case of finite covolume one is forced to consider the
cohomology of the subgroup with coefficients in the distribution vector globalization. This leads to problems in the analysis
on weighted function spaces on symmetric spaces.
These are already complicated in the rank-one case. In
Resolutions of distribution globalizations of Harish-Chandra modules
and cohomology .
we show finiteness and relate the cohomology to automorphic forms.
To obtain the corresponding results in the higher rank (as in the cocompact case) we rely on deep results of Kashiwara/Schmid and Franke
Cohomology of S-arithmetic groups in globalizations of Harish-Chandra modules and automorphic forms.
Open problems :
- Show the exactness of the resolutions using Ehrenpreis' principle and Paley Wiener theorems!
- explicit computations
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