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Let π be a unitary irreducible representation of almost connected solvable p -adic group G ( p ≠ 2 ) and Ω the attached coadjoint orbit according to Duflo's orbit method. We suppose Ω is closed. Then the representation is admissible and we prove a character formula expressing in some neighborhoods of each semi-simple element s of G the character of π in terms of the Fourier transform of the Liouville measure on the set of s -fixed points in Ω . As a by-product we establish in Maktouf (2012) [14] a character formula in the general case and derive the description of the Plancherel formula. [ABSTRACT FROM AUTHOR]

Abstract: Following Duflo–Vergne (for real reductive groups) and Khalgui–Torasso (for real almost algebraic groups), we prove a Plancherel formula for almost connected solvable p-adic groups (). To do this, we give a necessary and sufficient condition for a unitary irreducible representation to be admissible in terms of Kirillov–Duflo orbit method. Then we establish a character formula around each semi-simple element. [Copyright &y& Elsevier]