Slowly-increasing functions

Automorphic representations satisfy a growth condition which we may as well factor out into a separate definition.

We define the following temporary “size” function \(s:\operatorname{GL}_n(\mathbb{R})\to\mathbb{R}\) by \(s(M)=trace(MM^T+M^{-1}M^{-T})\) where \(M^{-T}\) denotes inverse-transpose. Note that \(s(M)\) is always positive, and is large if \(M\) has a very large or very small (in absolute value) eigenvalue.

DefinitionAutomorphicForm.GLn.IsSlowlyIncreasing #

We say that a function \(f:\operatorname{GL}_n(\mathbb{R})\to\mathbb{C}\) is slowly-increasing if there's some real constant \(C\) and positive integer \(n\) such that \(|f(M)|\leq Cs(M)^n\) for all \(M\in\operatorname{GL}_n(\mathbb{R})\).