On limit multiplicities of representations with cohomology in the cuspidal spectrum (Q580516)

Let G be a connected noncompact semisimple Lie group with a non-empty set \(G_ d\) of discrete series representations, and let V be an irreducible finite-dimensional representation of G. Given a fixed invariant measure v of G the formal degree of a discrete series representation \(w\in G_ d\) with respect to v is denoted by \(d_ w\). For a tower \(\{\Gamma_ i\}_{i\in {\mathbb{N}}}\) of cocompact discrete subgroups of G \textit{D. L. DeGeorge} and \textit{N. R. Wallach} [Ann. Math., II. Ser. 107, 133-150 (1978; Zbl 0397.22007)] have proved that \[ \lim_{i\to \infty}m(w,\Gamma_ i)/v(\Gamma_ i\setminus G)= \begin{cases} d_ w, &\text{ if \(w\in G_ d,\)} \\ 0, &\text{ if \(w\not\in G_ d\)}\end{cases} \tag{*} \] where \(m(w,\Gamma_ i)\) denotes the multiplicity with which an irreducible unitary representation of G occurs in the cuspidal spectrum \(L^ 2_ 0(\Gamma_ i\setminus G)\) \((=L^ 2(\Gamma_ i\setminus G)\) in this case). In the paper under review the more general case of a tower of arithmetic subgroups of finite covolume is dealt with. By purely cohomological methods it is proved that one has (for V and thus w regular) \[ | \hat G_ d(V)|^{-1}\lim_{i\to \infty}\sum_{w}m(w,\Gamma_ i)/v(\Gamma_ i\setminus G)=d_ w \] where one sums over the set \(\hat G_ d(V)\) of all discrete series representations whose infinitesimal character coincides with the one of the representation contragredient to V (all \(w\in \hat G_ d(V)\) have the same formal degree \(d_ w).\) Taking up the methods of DeGeorge-Wallach the formula (*) is proved in general by now by \textit{G. Savin} (Limit multiplicities for discrete series repesentations, preprint 1987) using in its final step the result of the authors described above.