Potential Theory on Non-Unimodular Groups

Abstract

Let G be a connected Lie group and let X ₁,... X _k be left invariant fields (i.e. X f _g = (X f) _g , f _g (x) = f(gx)) that generate the Lie algebra. We can consider then $\Delta = - X_1^2 - X_2^2 \cdots - X_k^2$ and T _t = e^−tΔ which is a convolution semigroup since it commutes with the left action of G. It follows that if we denote by d ^ℓ g the left Haar measure of G, we have $T_t f(x) = \int\limits_G {f(y)\phi _t (y^{ - 1} x)d^l y}$, cf. [1]. The behaviour of ϕ _t as t → ∞ when G is unimodular, i.e. when D ^ℓ g = dg (= the right Haar measure up to multiplicative constant), is well understood, cf. [1], [2].