Let \(k\) be a local field of characteristic zero. Let \(G\) be the \(k\)-points of a connected linear algebraic group defined over \(k\). If \(k= \mathbb{R}\) is the field of real numbers, we also allow \(G\) to be a real Lie group. Let \(B\) be a minimal parabolic subgroup containing a maximally \(k\)-split torus \(D\). We pick a positive root system \(\Phi^+\) of \(G\) with respect to \((B,D)\) and let \(\{\alpha,\dots,\alpha_n\}\) denote its simple roots. Let \(\delta_B\) denote its modular function. Let \(K\) be a good maximal compact subgroup of \(G\).
Let \((\rho,V)\) be a rational representation of \(G\) over a finite-dimensional \(k\)-vector space \(V\). We assume that \(V\) is an excellent representation of \(G\), i.e. the fixed points \(V^{G_i}=0\) for every non-compact simple factor \(G_i\) of \(G\). In order to simplify the notations, we also assume that \(V\) is an irreducible representation of \(G\) in this review. Let \(\lambda\) and \(\rho\) be the highest and lowest weights of \(V\) with respect to \(D\). Finally we form the group \(\ltimes_\rho V\).
Let \((\Pi,W)\) be a unitary representation of \(G\ltimes V\). This paper only considers such \(\Pi\) that the set of \(V\)-fixed vectors is zero, i.e. \(W^V=0\). The main result is to give bounds on the rate of growth of the matrix coefficients of \(K\)-finite vectors in \(\Pi\).
In order to describe the main theorem, we need to introduce more notations. Let \(r\) denote the number of roots appearing in \(V\). Let \(q_1= 3^{1-r}\) if \(\dim V>1\) and \(q_1= 3^{2-r}\) if \(\dim V=1\). We write \(\delta_B= \sum^n_{i=1} d_i\alpha_i\) and \({q\over 2}(\lambda-\rho)= \sum^n_{i=1} e_i\alpha_i\) as rational linear combinations of simple roots. Let \(m\) be an integer such that \(2m\geq\max_{1\leq i\leq n}\{{d_i\over e_i}\}\). Let \(\Xi_G\) denote the Harish-Chandra Xi-function. It is a bi-\(K\)-invariant function on \(G\). The main Theorem 1.4 states that \((\Pi,W)\) is \((K,\Xi^{{1\over m}}_G)\) bounded on \(G\), i.e. for a pair of \(K\)-finite vectors \(v,w\in W\), we have \[ \langle\Pi(g)v,w\rangle\leq \sqrt{\text{dim\,Span}\{Kv\}} \sqrt{\text{dim\,Span}\{Kw\}} \Xi^{{1\over m}}_G(g) \] for all \(g\in G\).
The author proves the main theorem by first showing that \(\Pi\) is strong \(L^{2m+\varepsilon}(G)\) and then applying a result by M. Cowling et al. [J. Reine Angew. Math. 387, 97–110 (1988; Zbl 0638.22004)] which implies that the representation is \((K,\Xi^{{1\over m}}_G)\) bounded on \(G\). By modifying the proofs for various specific situations, the author is able to obtain better bounds if \(G= \text{SL}_3\) or \(G\) has \(k\)-split rank 1.