Glasnik Matematicki, Vol. 32, No.2 (1997), 179-199.


Dragan Milicic and Pavle Pandzic

Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract.   Let G0 be a connected real semisimple Lie group with finite center. Let K0 be a maximal compact subgroup of G0. Denote by g the complexification of the Lie algebra of G0 and by K the complexification of K0. Let T be a complex torus in K and t its Lie algebra. We can consider the categories M(g,K) and M(g,T) of Harish-Chandra modules for the pairs (g,K) and (g,T). Clearly, M(g,T) is a full subcategory of M(g) and M(g,K) is a full subcategory of M(g,T). The natural forgetful functors have right adjoints GammaT, GammaK and GammaK,T. These adjoints are called the Zuckerman functors. Zuckerman functors are left exact and have finite right cohomological dimension. Therefore, one can consider their right derived functors. They are related by the obvious Grothendieck spectral sequence

Rp GammaK,T (Rq GammaT(V)) --> Rp+q GammaK(V),

for any V in M(g). In this paper, this spectral sequence is investigated in the case where V is the Verma module M(lambda). The main result is the decomposition formula which strengthens the Duflo-Vergne formula. This decomposition formula implies in particular that the above spectral sequence degenerates.

The authors first show that a bounded complex (satisfying certain finiteness conditions) is isomorphic to the direct sum of its cohomologies if and only if its endomorphism algebra has maximal possible dimension. Next, they analyze the derived functors of the right adjoint to a forgetful functor. In the final section, they prove a decomposition formula for derived Zuckerman functors for tori, which then leads to their main results.

1991 Mathematics Subject Classification.   22E46.

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