Spectral multiplicity of derived functor modules
Let \(\Gamma\) be an arithmetic subgroup of a semisimple Lie group G. Each irreducible unitary representation \(\pi\) of G occurs with a finite multiplicity \(m_{\Gamma}(\pi)\) in the G-module \(L^ 2(\Gamma \setminus G)\). In his previous paper [Am. J. Math. 107, 367-376 (1985; Zbl 0569.22010)] the author gave a proof of a conjecture of Langlands which gave the multiplicities \(m_{\Gamma}(\pi)\) for \(\pi\) square integrable a cohomological interpretation as follows: Such a \(\pi\) is given by a character \(\lambda\) of a compact Cartan subgroup H of G. A character \(\lambda\) of H defines a line bundle \(L_{\lambda}\) over G/H on which \(\Gamma\) acts. The conjecture of Langlands states that the \(L^ 2\)- cohomology of \(\Gamma \setminus L_{\lambda}\) lives in a single degree and its dimension equals \(m_{\Gamma}(\pi).\) The paper under consideration generalizes the above to certain derived functor modules in the sense of Zuckerman which are given by a character \(\lambda\) as above. It is shown that under mild restrictive conditions on \(\lambda\) the \(L^ 2\)-cohomology of \(\Gamma \setminus L_{\lambda}\) lives in a single degree and its dimension equals \(m_{\Gamma}(\pi)\) for \(\pi\) the Zuckerman module attached to \(\pi\). As a byproduct we learn that the \(L^ 2\)-cohomology of \(\Gamma \setminus L_{\lambda}\) is a sum of nilpotent Lie algebra cohomologies and that these have \(m_{\Gamma}(\pi)\) as their dimension.
scientific article; zbMATH DE number 4212340
scientific article