Can one hear the shape of a Lie-type geometry?

In a celebrated paper 'Can one hear the shape of a drum?' (Kac 1966 Am. Math. Mon. 73 1-23) Kac asked his famous question on the consequences of isospectrality, which was eventually answered negatively in Gordon et al ( 1992 Invent. Math. 110 1-22) by the construction of non-congruent planar isospectral pairs. In this paper, we pose a similar question for Lie-type geometries, give partial answers and show that the question can be reduced to an important geometrical problem.