The Martin Compactification X ∪ ∂ X(λ)

Abstract

Motivated by the study of Brownian motion, Dynkin [D4] was the first to investigate the positive eigenfunctions and corresponding Martin boundaries for the Laplace—Beltrami operator on a symmetric space of non-compact type. He restricted his attention to the space SL(n,C)/SU(n). This space is especially amenable to a study of the Martin compactification because one has an explicit formula for the Green function Gx that is a consequence of a remarkable formula of Weyl — namely,$\prod\nolimits_{\alpha > 0} {\left( {{e^{\alpha \left( H \right)}} - {e^{ - \alpha \left( H \right)}}} \right)} = \sum\nolimits_{s \in W} {\left( {\det s} \right){e^{s \cdot p\left( H \right)}},}$ see Freudenthal [Fl, Proposition 47.14] — that is true for complex Lie algebras. Later, Nolde [N] published a note announcing the same results as Dynkin for any semisimple Lie group whose Lie algebra is a complex Lie algebra. This was followed by a note of 0l-shanetsky [01] that stated asymptotic formulas for the Green function and deduced, as a result, the results analogous to those of Dynkin and Nolde. While the proofs of Olshanetsky’s asymptotic formulas were recently published [02], they are insufficient, as pointed out in footnote 6 in Chapter I, to deduce the (correct) results given in [02] about the Martin compactification for ⋋ < ⋋ o.