Perturbation theory for twisted automorphic functions
Let \(\Gamma\) be a cofinite non-cocompact discrete subgroup of \(\text{SL}(2,\mathbb{R})\) and \(\rho\) a finite-dimensional irreducible unitary representation of \(\Gamma\) acting on a \(d\)-dimensional unitary vector space. \((\Gamma,\rho)\) is called essentially cuspidal if the cusp form eigenvalues for \((\Gamma,\rho)\) embedded in the continuous spectrum satisfy Weyl's law. The purpose of the paper under review is to show that the generic \((\Gamma,\rho)\) is not essentially cuspidal if one assumes for the trivial representation that the dimension of the eigenspaces containing at least one newform is bounded. The present paper extends previous work of \textit{R. Phillips} and \textit{P. Sarnak} [Geom. Funct. Anal. 4, 93-118 (1994; Zbl 0804.11038)] to cover the case of multidimensional irreducible unitary representations under the premise quoted above. To simplify the exposition the author limits himself to the case \(\Gamma=\Gamma(2)\), the principal congruence subgroup of level 2 in \(\text{SL}_2(\mathbb{Z})\). The author's approach to the problem is based on perturbation theory.
multidimensional irreducible unitary representations
scientific article; zbMATH DE number 1000117