Noninvariant integrals on semisimple groups of \(\mathbb R\)-rank one (Q1066286)

Let \(G\) be a connected semisimple Lie group with finite center, \(G_c\) its simply connected complexification, \(K\) a maximal compact subgroup of \(G\), and: \(G\subset G_c\), \(\text{rank}\, G=\text{rank}\, K\), \(\text{rank}\, G/K=1\). Let \(A\) be a Cartan subgroup of \(G\), \(a\) a regular element in \(A\), \(v_A(x)\) for \(x\in G\) a weighting function on \(G\). The weighted orbital (noninvariant) integral on \(G\) is defined by \[ T^A_f(a)=\varepsilon^A_R(a) \Delta_A(a)\cdot \int_{G/A} f(xax^{-1})\cdot v_A(x)\,d_G(x). \] The aim of the paper is to study noninvariant integrals on \(G\): asymptotics, the Fourier transformation, and so on.