Algebras of invariant functions on the Shilov boundaries of Siegel domains

Let D = G / K D=G/K be a bounded symmetric domain and K / L K/L the Shilov boundary of D D . Let N \mathcal {N} be the Shilov boundary of the Siegel domain realization of G / K G/K . We consider the case when D D is the exceptional non-tube type domain of the type ( e 6 ( − 14 ) , s o ( 10 ) × s o ( 2 ) ) (\mathfrak {e}_{6(-14)}, \mathfrak {so}(10)\times \mathfrak {so}(2)) . We prove that ( N ⋊ L , L ) (\mathcal {N}\rtimes L, L) is not a Gelfand pair and thus resolve an open question of G. Carcano.