Decay of matrix coefficients on reductive homogeneous spaces of spherical type (Q463973)

Let \(Z\) be a homogeneous space \(Z = G/H\) of a real reductive Lie group \(G\) with a reductive subgroup \(H.\) The authors define \(Z\) to be of spherical type if there exists a minimal parabolic subgroup \(P\) such that \(PH\) is open in \(G.\) For a continuous linear representation \((\pi, E)\) of \(G,\) on a Banach space \(E,\) and for a vector \(v \in E\) and \(\eta \in E^\ast\) define the matrix coefficient \[ m_{v, \eta}(g) = \eta(\pi(g^{-1}v) \quad g \in G. \] The paper is concerned with estimates for these matrix coefficients. Assume that \(Z\) is of spherical type and \(E = V^\infty\) is the smooth globalization of a Harish-Chandra module \(V.\) Let \(K\) be a maximal compact subgroup, \(P= MAN\) the Langlands decomposition of the minimal parabolic subgroup and \(A^{+}\) the Weyl chamber associated to \(P.\) Then, for every \(v \in V\) there is a constant \(C > 0\) such that \[ |m_{v, \eta}(g)| \leq C~a^{\Lambda_V} (1+\| \log a\|)^{d_V}, \quad a \in A^{+}, \] where \(\Lambda_V \in \mathfrak{a}^\ast \) and \(d_V \in \mathbb N\) are determined by \(V.\) Several examples are also discussed.