Commutative spaces which are not weakly symmetric

JORGE LAURET

BULLETIN OF THE LONDON MATHEMATICAL SOCIETY

OXFORD UNIV PRESS

Lugar: Oxford; Año: 1998 vol. 30 p. 29 - 36

0024-6093

In 1956 A. Selberg introduced weakly symmetric spaces in theframework of his development of the trace formula and proved that ina weakly symmetric space the algebra of all invariant (with respectto the full isometry group) differential operators is commutative(\cite{S}).  In this paper Selberg asks whether the converse holds.In the present work we shall answer this question by presentingexamplesof commutative spaces which are not weakly symmetric.  These examplesarise in the quaternionic analogues to the Heisenberg group, endowedwith certain special metrics (see Theorem \ref{contra} and theexplicit realization after it)\footnote{1991 {\it Mathematics SubjectClassification} 22E30, 53C30, 22E25, 43A20. }.