Abstract. Multivariable extensions of the Scott Brown technique are used to prove that on a quite general class of domains $D$ in ${\msbm C}^n$, or in a complex submanifold of ${\msbm C}^n$, each subnormal tuple $T\in L(H)^n$ with an isometric weak$^*$ continuous $H^{\infty }(D)$-functional calculus is {\it reflexive}. To obtain our results we use methods developed by Aleksandrov in his abstract approach to the inner function problem, and we prove new results for Henkin measures on suitable complex domains.

AMS Subject Classification (1991): 47A13, 47A15, 47A60, 47B20, 47L45

Received December 28, 2004. (Registered under 5895/2009.)