Analysis on Lie groups with polynomial growth
A Lie group is called a Lie group of polynomial growth, if the volume of a ball does not grow faster than a power of its radius. All Lie groups of polynomial growth are unimodular. Compact and nilpotent Lie groups have polynomial growth. Each non-compact semisimple Lie group has exponential growth (the volume of a ball grows exponentially with its radius). In the book under review the authors analyze the structure of connected Lie groups of polynomial growth -- with particular emphasis on global properties. The global properties of a manifold are intimately related to the asymptotics of the heat equation. Each connected simply connected Lie group \(G\) is the semidirect product of a semisimple Lie group \(M\) acting on a solvable Lie group \(Q\), the radical of \(G\). If \(G\) has polynomial growth, then \(M\) is compact and \(Q\) also has polynomial growth. This group \(Q\) can be obtained from a nilpotent group \(Q_N\) by modification of the group product. The group \(Q_N\) is called the nilshadow of \(Q\). The groups \(Q\) and \(Q_N\) are identical as manifolds but differ in their group structure. So, the group \(G\) can be identified with the ``cylindrical manifold'' \(M\times Q_N\) equipped with a new product. Then the asymptotic evolution of solutions of the heat equation is in the direction of the nilshadow \(Q_N\). With each right-invariant subelliptic operator \(H\) of second order on \(G\) one can associate a similar operator \(\widehat H\) on \(Q_N\) which governs the asymptotic behaviour. Let \(K\) and \(\widehat K\) on \(G\) and \(Q_N\), respectively. The kernel \(\widehat K\) has Gaussian behaviour and this \(\widehat K\) describes the asymptotic behaviour of \(K\).
scientific article; zbMATH DE number 2010202