Dimensional gap in semisimple compact Lie groups via Fourier series (Q393421)

For a metric group \(G \), one can ask for the possible Hausdorff dimensions of a measurable Borel subgroup. The author has shown in [Mathematika 59, No. 2, 497--511 (2013; Zbl 1272.22006)] that for any connected nilpotent Lie group, one can find Borel subgroups of any Hausdorff dimension between 0 and dim\((G )\). In this paper the author considers real compact semisimple Lie groups. Let \(d \) be the dimension of \(G \), let \(\Delta \) be a root system of \(G \). The author shows that there exists a number \(\alpha_0\) which has the property that there exists a constant \(C>0 \) such that for any irreducible representation \((\pi_\lambda, V_\lambda) \) of \(G \) with highest weight \(\lambda \) one has that \(\|\lambda\| ^{d-\alpha_0}\leq C\text{dim}V_\lambda \). To find such a number \(\alpha_0 \), let \(p \) be the maximal number of elements in \(\Delta \) contained in a hyperplane and one can take \(\alpha_0:=d-(\frac{| \Delta|-p}{2}) \). Using the Plancherel formula the author shows that if \(\mu \) is a finite Borel measure with Hölder exponent \(\alpha >\alpha_0 \), then \(\mu^{\ast k} \) is absolutely continuous with respect to Haar measure if \(k>\frac {d-\alpha_0}{\alpha-\alpha_0} \). It follows from this estimate that there is a dimension gap, i.e. the real compact Lie group has no strict Borel subgroup of Hausdorff dimension larger than \(\alpha_0 \). Using again the Plancherel formula, the author can show that for any two Borel subsets \(A,B \) of \(G \) one has that \(\text{dim}_H(AB)\geq \min\{d, \text{dim}_H(A)+\text{dim}_H(B)-\alpha_0\} \). In the second part of the paper, the author proves similar results for the groups \(\mathrm{SL}_{r+1}(\mathbb Z_p) \) , based on an inequality of the type \(p ^{rn}\leq C \text{dim}(V_\pi)\), for generic irreducible representations \(\pi \) of the groups \(\mathrm{SL}_{ r + 1} (\mathbb Z/p^n\mathbb Z)\).