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G: itself. Suppose that r l < i 5 r,-then, in virtue of the theory of GarlandRaghunathan, the percuspidal subgroups of the pair (GT, r:) are but certain minimal parabolic subgroups of GT,the totality of which is finite, modulo r? conjugacy. Suppose that r z < i 5 r-then the percuspidal subgroups of the pair (G:, r:) are the same as those of the pair (GT, ZT . r:), where Z y is the (finite) center of G:, and the latter are simply the pullbacks to G: of the percuspidal subgroups of the pair (Ad(G:), Ad(T:)).

1 . Z . G,,,, n G,,,, = r,. To finish up, then, it suffices to show that N’ n 1 is a lattice in N‘. Thus, since ( P o , S o ) is a 1-percuspidal split parabolic subgroup of G and (P‘, S’) is a dominant successor of (Po, So), we conclude that indeed N‘ n 1 is a lattice in N ‘ , as desired. // LATTICES IN A REDUCTIVE LIE GROUP 43 Remark Let ( P , S) be a split parabolic subgroup of G such that (i) r n P is contained in S , (ii) N / N n r is compact, (iii) S/S n r is of finite volume. Then (P, S) is a r-cuspidal split parabolic subgroup of G.

M. S. OSBORNE AND G. WARNER: The Selberg trace formula I: r-rank one lattices (to appear). M. S. OSBORNE AND G. WARNER: The Selberg trace formula 11: Partition, reduction, truncation (to appear). ] Let us give one final reference, in essence a numerical example, albeit instructive : A. B. VENKOV:The Selberg trace formula for SL(3, Z), J . Soviet Math. 12( 1979), 384-424. As a blanket convention, any reference to Harish-Chandra or Langlands in the text proper is understood to be to that author’s respective Springer Lecture Notes cited above.

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