Limit multiplicities of cusp forms (Q1120687)

Let G be a connected non compact 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 \[ (*)\quad \lim_{i\to \infty}\frac{m(w,\Gamma_ i)}{v(\Gamma_ i\setminus G)}=\begin{cases} d_ w,&\quad w\in G_ d\\ 0,&\quad w\not\in G_ d\end{cases} \] 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 congruence subgroups of finite covolume is dealt with. Taking up the methods of DeGeorge-Wallach the formula (*) is proved now for any irreducible unitary representation w of G occuring with multiplicity \(m(w,\Gamma_ i)\) in the space of cusp forms \(L^ 2_ 0(\Gamma_ i\setminus G)\). Note that when \(\pi\) is not a discrete series representation then the limit multiplicity in zero. In the final step of the argument results of \textit{J. Rohlfs} and \textit{B. Speh} [Duke Math. J. 55, 199-211 (1987; Zbl 0626.22008)] are used.