Formule de Plancherel sur $GL_n \times GL_n \backslash GL_{2n}$

TL;DR

This paper establishes an explicit Plancherel formula for the quotient space involving the group $GL_{2n}$ over a non-archimedean local field, utilizing Jacquet-Shalika zeta functions, and derives related decompositions.

Contribution

It provides a new explicit Plancherel formula for $H_n(F) ackslash GL_{2n}(F)$ using Jacquet-Shalika theory, advancing harmonic analysis on these symmetric spaces.

Findings

Explicit Plancherel formula derived for the quotient space.

Decomposition results for $H_n(F) ackslash GL_{2n}(F)$ and $GL_{n}(F) imes GL_{n}(F) ackslash GL_{2n}(F)$.

Application of Jacquet-Shalika zeta functions to harmonic analysis.

Abstract

Let $F$ be a non archimedian local field and $H_n(F)$ the Shalika subgroup of $GL_{2n}(F)$. We prove an explicit Plancherel formula for $H_n(F) \backslash GL_{2n}(F)$ using the theory of Jacquet-Shalika of zeta functions and we deduce Plancherel decompositions for $H_n(F) \backslash GL_{2n}(F)$ and $GL_{n}(F) \times GL_{n}(F) \backslash GL_{2n}(F)$.