Intersection numbers of special cycles

Let G /Q be a connected semisimple algebraic group defined over Q, and let a, r be two Q-rational automorphisms of finite order of G which commute with each other. We assume that the group G(R) of real points of G is noncompact. Let i be a torsionfree arithmetic subgroup of G(Q) which is stable under the group (a, r) generated by a and r. Then a and r act on the space X of maximal compact subgroups of G(R) and on the locally symmetric space X/I in a natural way. If G(f..l) denotes the reductive algebraic Q-group of fixed points of f..l = a , r acting on G, then the corresponding symmetric space X(f..l) is acted upon by the arithmetic group 1(f..l) of fixed points of f..l acting on i. These give rise to nonempty closed immersed submanifolds C(a) = X(a)/1(a) and C(r) = X(r)/1(r) of X/i. They are called special cycles. We assume that X/i, C(a), and C ( r) are oriented and that the intersection C (a) n C ( r) is compact. Then, if dim C(a) + dim C(r) = dim X, the intersection number