Bounds on multiplicities of symmetric pairs of finite groups (Q6603916)

Let \(G\) be a group and let \(X\) be a transitive \(G\)-space. A problem of representation theory is to compute the multiplicities with which irreducible representations of \(G\) appear in the space of functions on \(X\). In the algebraic setting, if \(G\) is a connected reductive algebraic group over \(\mathbb{C}\) and \(X\) is a spherical \(G\)-variety, then \(\mathbb{C}[X]\) is multiplicity-free as a \(G\)-representation. In non-algebraic settings, these multiplicities may be greater than one.\N\NLet \(\Gamma\) be a finite group, \(\theta\) an involution of \(\Gamma\) and \(\rho\) an irreducible complex representation of \(\Gamma\). In the paper under review, the authors bound \(\dim \rho^{\Gamma_{\theta}}\) in terms of the smallest dimension of a faithful \(\mathbb{F}_{p}\)-representation of \(\Gamma/O_{p}(\Gamma)\), where \(p\) is any odd prime and \(O_{p}(\Gamma)\) is the maximal normal \(p\)-subgroup of \(\Gamma\). As a consequence they deduce that if \(\mathbf{G}\) is a group scheme over \(\mathbb{Z}\) and \(\theta\) is an involution of \(\mathbf{G}\), then the multiplicity of any irreducible representation in \(C^{\infty} \big (\mathbf{G}(\mathbb{Z})_{p}) / \mathbf{G}^{\theta}(\mathbb{Z})_{p} \big )\) is bounded uniformly in \(p\).\N\NThis paper contains much more information on the same topic, however their formulation is too complicated to be reported here.