Explicit Plancherel Theorems for H(q1, q2) and SL2(F)

For each pair q1 >= q2 >= 1 of real numbers, the authors define a complex algebra (with identity) H = H(q1, q2), an associated involutive Banach algebra A = A(q1, q2) and its associated enveloping C*-algebra C*(A), and a quotient C*-algebra C-r*(A). Deformation arguments are used to obtain an explicit Plancherel formula for C-r*(A); the unitary dual of C-r*(A) is explicitly described, as is the unitary dual of C*(A). These results, together with the theory of types, are used to obtain the Plancherel measure for the group SL2(F), where F is a complete non archimedean local field with arbitrary residual characteristic p. This includes an explicit description of the reduced dual. The methods, but not the results, are independent of p.