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Of course the groups F, are then arithmetic subgroups of G. In this situation, it seems natural to assume that, as n-+ ~ , the spectral decomposition of L2(F,\G) will approximate that of G. In particular, let 3 be a discrete series representation of G. If F is a subgroup of G, let re(F, 6) be the multiplicity of 3 in the representation of G, by right translations, on L2(F\G). Choose a Haar measure dg on G; if F is arithmetic, let v(F\G) be the (finite) volume of F\G for the measure induced by dg. Let d(3) be the formal degree of 3 for the measure dg.

On limit multiplicites of discrete series representations in spaces of automorphic forms