The sup-norm problem on the Siegel modular space of rank two (Q2816470)

Let \(\mathrm{Sp}_4({\mathbb R})\) be the real symplectic group of rank \(2\) and let \(\Gamma=\mathrm{Sp}_4({\mathbb Z})\) be the Siegel modular group. The Riemannian symmetric space \(\mathrm{Sp}_4({\mathbb R})/\mathrm{Sp}_4({\mathbb Z})\) is the Siegel upper half space \({\mathcal H}\). The first result proved in the paper under review can be stated as follows.NEWLINENEWLINETheorem 1: There exists \(\delta>0\) such that for any compact subset \(\Omega\) of \(\Gamma\backslash {\mathcal H}\) and any \(L^2\)-normalized joint eigenfunction \(F\) in \(L^2(\Gamma\backslash{\mathcal H})\) with Laplace eigenvalue \(\lambda_F\) one has NEWLINE\[NEWLINE\| F|_{\Omega}\|_\infty\ll_\Omega(1+\lambda_F)^{1-\delta}.NEWLINE\]NEWLINE Next, let \(G\) be a real connected semisimple Lie group with finite center and \(K\) a maximal compact subgroup of \(G\). Fix an Iwasawa decomposition \(G=KAN\) and denote by \({\mathfrak a}\) the Lie algebra of \(A\). Fix a system of positive restricted \({\mathfrak a}\)-roots and write \(\rho\) for half their sum. If \(W\) is the Weyl group of \(G\), let \(C_\rho\) be the convex hull of the points \(w\cdot\rho\), \(w\in W\). The second result proved in the paper is then stated as:NEWLINENEWLINETheorem 2: Let \(B\subset{\mathfrak a}\) be a bounded subset. Then for any \(H\in B\) and any \(\xi=\lambda+i\eta\in{\mathfrak a}^*+iC_\rho\), the elementary spherical function \(\varphi_\xi\) of \(G\) with parameter \(\xi\) satisfies NEWLINE\[NEWLINE\varphi_\xi(\exp(H))\ll_B\Pi_j(1+\mid\mid\lambda_j\mid\mid\cdot\mid\mid H_j\mid\mid)^{-1/2},NEWLINE\]NEWLINE where \(\lambda_j\) and \(H_j\) denote the projections onto the simple factors of \({\mathfrak a}\) respectively \({\mathfrak a}^*\).