Spectral deformations of automorphic functions over graphs of groups
Let X be a space on which a group \(\Gamma\) acts isometrically, and let \(\Gamma '\) be a normal subgroup of finite index. Then under wide assumptions the space \(L^ 2(\Gamma '\setminus X)\) splits into the direct sum of the spaces \(L^ 2(\Gamma \setminus X,\rho)^{\deg \rho}\) where \(\rho\) ranges over the irreducible representations of \(\Gamma '\setminus \Gamma\). A fundamental question in the spectral theory of automorphic forms is then, whether \(L^ 2(\Gamma '\setminus X)\) is just what stems from \(L^ 2(\Gamma \setminus X)\) or whether the spaces \(L^ 2(\Gamma \setminus X,\rho)\) make an additional contribution and what this contribution may be. This problem was discussed by \textit{P. Sarnak} (cf. the preceding review), in the case of automorphic forms on hyperbolic spaces. In particular, Sarnak investigated the disappearance of discrete spectra under deformations and this made him conjecture that there should exist groups whose spectrum is abundant in \(L^ 2(\Gamma \setminus X)\) and is nevertheless destroyed by a deformation. The aim of the paper under review is to prove this conjecture in the context of the Bruhat-Tits building X attached to \(GL_ 2(K)\) where K is the completion at infinity of the function field of genus 0 over \({\mathbb{F}}_ q\). Let \(\Gamma\) be a discrete subgroup of the group of isometries of X such that \(\Gamma\setminus X\) is of finite volume. The square-integrable eigenfunctions of the invariant operators on X span a finite-dimensional space in \(L^ 2(\Gamma \setminus X)\). Neglecting the trivial two-dimensional part of this space generated by the constant and alternating functions let \(L^ 2(\Gamma \setminus X)_{dis}\) be the non-trivial part. Main Theorem: For every integer \(N>0\) there exists a group \(\Gamma =\Gamma_ N\), such that dim \(L^ 2(\Gamma \setminus X)_{dis}=N\), yet dim \(L^ 2(\Gamma \setminus X,\chi)_{dis}=0\) for every character \(\chi\neq 1\) in a neighbourhood of the identity character. Hence, in strong contrast to what is known about the spectral behaviour of congruence subgroups of the modular group, the author gives the following answer to the question raised above: For any positive integer N there exist arbitrarily large integers M and pairs \(\Gamma '<\Gamma\) with dim \(L^ 2(\Gamma \setminus X)_{dis}=N\) such that \([\Gamma:\Gamma ']=M\) yet \(L^ 2(\Gamma '\setminus X)_{dis}=L^ 2(\Gamma \setminus X)_{dis}\).
scientific article; zbMATH DE number 4160850
scientific article