Let K be a locally compact,
non-Archimedean field of residual characteristic p, and let D be a central division
algebra of dimension n2 over K. In constructing the irreducible unitary
representations of D×, a technical question repeatedly arises. Let x ∈ D, and let
x1 be “close” to x (in the sense that, for the usual absolute value on D,
|x − x1| < |x|). Let Dx, Dx1 be the subalgebras of elements commuting with x, x1
respectively. Is it possible to pick a prime element η1∈ Dx1 and an element
η0∈ Dx that are also close, and how close can η, η1 be to one another? The
first part of this paper analyzes this problem. It turns out that η, η1 can be
chosen close enough to one another so that Clifford-Mackey theory easily
permits the construction of (Dx)∧ only if p2= n or p2|n. The construction has
been given in earlier papers except for the case where p|n, p≠n, and p2|n;
the second part of the paper is a construction of (Dx)∧ in this remaining
case.
