Non-minimal modular symbols for GL(n)
Let M be a Levi factor of a \({\mathbb{Q}}\)-parabolic subgroup P of Gl(n). Let X be the symmetric space for Gl(n) with base point e, \(\Gamma\) an arithmetic subgroup of Gl(n,\({\mathbb{Q}})\), and g any element of Gl(n,\({\mathbb{Q}})\). Then the projection of gM(\({\mathbb{R}})e\) in \(\Gamma\) \(\setminus X\) is a closed submanifold. Its fundamental class defines by Lefschetz duality a cohomology class in \(H^*(\Gamma \setminus X,C)\). This class is called a modular symbol of type M for \(\Gamma\). The main theorem states: Suppose that \(n>1\). Let \(\Gamma\) be a full congruence subgroup of level N, \(N\geq 1\), p a prime that divides N precisely once, and M a Levi factor of a parabolic subgroup with one block of size greater than n/2. Then the span of the modular symbols of M contains no irreducible f-cuspidal representation of SL(n,\({\mathbb{Z}}/p)\).
scientific article; zbMATH DE number 4097680
scientific article