Bounds on smooth matrix coefficients on \(L^{2}\)-spaces (Q1039325)

Let \(G\) be a semisimple Lie group with finitely many connected components and with finite center. Let \(K\) be a maximal compact subgroup of \(G\). Let \(G=KAN\) be an Iwasawa decomposition. Denote the Lie algebra of \(A\) by \(\mathfrak{a}\), and its complexification by \(\mathfrak{a}_\mathbb{C}\). Identify \(\mathfrak{a}\) with \(\mathfrak{a}^*\) via the Killing form. Let \(\mathfrak{a}^+\) be the Weyl chamber corresponding to the system of positive roots for \((\mathfrak{g},\mathfrak{a})\) determined by \(N\). One says that \(\lambda\in\mathfrak{a}_\mathbb{C}^*\) is dominated by \(\lambda'\in\mathfrak{a}_\mathbb{C}^*\) if the real part of \(\lambda'-\lambda\) is nonnegative on \(\mathfrak{a}^+\). Let \(X\) be a smooth \(G\)-space with a \(G\)-invariant measure. Then \(L^2(X)\) is a unitary representation of \(G\). Let \(\widehat G\) be the unitary dual of \(G\) equipped with the Fell topology. The representation \(L^2(X)\) can be decomposed into irreducibles via a direct integral over \(\widehat G\) with respect to a Borel measure. In particular, one can consider the support \(\text{supp}(L^2(X))\) of \(L^2(X)\). Let \(\widehat G_K\) be the spherical unitary dual of \(G\), identified with a closed subset of the dominant Weyl chamber in \(\mathfrak{a}_\mathbb{C}^*\). The main result of the paper under review gives certain upper bounds for \(K\)-finite and \(\mathfrak{k}\)-smooth matrix coefficients of representations weakly contained in \(L^2(X)\), under the assumption that \(\text{supp}(L^2(X))\cap \widehat G_K\), considered as a subset of the dominant Weyl chamber in \(\mathfrak{a}_\mathbb{C}^*\), is dominated by a real \(\lambda_0\). This generalizes a result of Cowling, Haagerup and Howe. The result is applied in the example of the \(O(p,q)\)-representation \(L^2(\mathbb{R}^{p+q})\).