# Plancherel measure for GL(n,F) and GL(m,D): Explicit formulas and Bernstein decomposition

Aubert, Anne-Marie and Plymen, Roger (2005) Plancherel measure for GL(n,F) and GL(m,D): Explicit formulas and Bernstein decomposition. Journal of Number Theory, 112. pp. 26-66. ISSN 0022-314X

Let F be a nonarchimedean local field, let D be a division algebra over F, let GL(n) = GL(n,F). Let $\nu$ denote Plancherel measure for GL(n). Let $\Omega$ be a component in the Bernstein variety $\Omega(\GL(n))$. Then $\Omega$ yields its fundamental invariants: the cardinality q of the residue field of F, the sizes m_1,..., m_t, exponents e_1,...,e_t, torsion numbers r_1,...,r_t$, formal degrees d_1,...,d_t and conductors f_{11},..., f_{tt}. We provide explicit formulas for the Bernstein component$\nu_{\Omega}$of Plancherel measure in terms of the fundamental nvariants. We prove a transfer-of-measure formula for$\GL(n)\$ and establish some new formal degree formulas. We derive, via the Jacquet-Langlands correspondence, the explicit Plancherel formula for GL(m,D).