Invent. math. 83, 383-401 (1986) Irl verl tiorl es mathematicae Springer-Verlag 1986 On liftings and cusp cohomology of arithmetic groups J.-P. Labesse 1 and J. Schwermer 2 1 D~pt. de Math6matiques, Universit6 de Dijon, B.P. n ~ 138, F-21004 Dijon Cedex, France 2 Mathematisches Institut der Universit~it, Wegelerstr. 10, D-5300 Bonn 1, Federal Republic of Germany w 0. Introduction Let Fc G(E) be an arithmetic subgroup of a semi simple connected algebraic group G defined over an algebraic number field E, and let V be a finite dimensional complex representation of the group G~ = G(N| of real points of G. Then the cusp cohomology it/cusp(/", V) is a subspace of the cohomology of F with coefficients in V defined in analytical terms [3]; using the identifi- cation of the cohomology H*(F, V) of F with the relative Lie algebra cohomol- ogy it is given as Hc*usp(F, V) = H*(g, Ko~; L20(F\G 0o) | V) (1) and, in particular, it isolates a finite set (depending on V) of representations n of G o occurring in the cuspidal spectrum L20(F\G~) with finite multiplicities re(n, F). The study of the cusp cohomology of F is therefore the study of special types of cuspidal automorphic forms. For example, if G=SL2/~, FcSL2(~. ) and V= V k is of dimension k, then (1) is the Eichler-Shimura isomorphism 1 H~usp(r, Vk) ~- S~+ 1 (r) | s~+ , (r) where S~+I(F ) is the space of holomorphic or anti-holomorphic cusp forms with respect to F of weight k+ 1; and the dimension of S~+ I(F) is the multi- plicity m(D~,F) of the holomorphic (antiholomorphic) discrete series represen- tation D~ of SL2(~ ). More generally, if G~o has discrete series representations the Selberg trace formula allows one to get hold of cusp cohomology classes by computing Euler-Poincar6 characteristics (cf. w But if G~ has no discrete series (e.g. in case G =SL, with n > 2 or G~--H(~) is a complex Lie group) the Euler-Poincar6 characteristics of the representations occurring in (1) vanish and the ordinary trace formula does not seem to be of any direct help. On the other hand, there are geometrical methods as additional tools. For example, if F~SL2((DE), (~s a ring of imaginary quadratic integers, or F