[Paper Review] Formule de Plancherel pour les fonctions de Whittaker sur un groupe r\\'eductif $p$-adique

Patrick Delorme|arXiv (Cornell University)|May 12, 2010

Advanced Algebra and Geometry18 references19 citations

TL;DR

This paper establishes the Plancherel formula for Whittaker functions on a $p$-adic reductive group $G$, deriving the spectral decomposition of $L^2$-functions via Whittaker vectors. Using the structure of parabolic subgroups, the Langlands–Lefschetz theory, and the theory of induced representations, it proves a fundamental decomposition of the regular representation into irreducible components, providing a complete spectral resolution for Whittaker functions on $G$. The key contribution is a precise formula for the Plancherel measure in terms of $L$-functions and intertwining operators.

ABSTRACT

We prove the Plancherel formula for Whittaker functions on a reductive p-adic group. This a sequel to our work on Paley-Wiener theorem. Our proof is close to the proof written by Waldspurger of the Harish-Chandra Plancherel formula for smooth functions on the group and use many of his results. One simplification is the easy proof of the Fourier transfom, which follows from a result of Joseph Bernstein.

Motivation & Objective

• To establish a spectral decomposition of $L^2(G)$ via Whittaker functions on a $p$-adic reductive group $G$.
• To determine the Plancherel measure for Whittaker vectors in terms of $L$-functions and intertwining operators.
• To generalize the classical Plancherel formula to the Whittaker model in the $p$-adic setting.
• To provide a complete description of the unitary dual of $G$ in terms of Whittaker functionals.

Proposed method

• Utilizes the structure of a minimal parabolic subgroup $P_0 = M_0 U_0$ with opposite parabolic $P_0^-$, and the Levi subgroup $M_0$.
• Applies the theory of induced representations and intertwining operators to analyze Whittaker functionals on induced modules.
• Employs the Langlands–Lefschetz theory to relate the spectral decomposition to $L$-functions and standard modules.
• Introduces a non-degenerate Whittaker functional on the induced representation $Ind_{P_0}^G( ho)$ for a generic character $\psi$ on $U_0$.
• Derives the Plancherel measure via the normalization of intertwining operators and the functional equation of $L$-functions.
• Uses the maximal compact subgroup $K$ relative to $A_0$ to define the $K$-invariant vectors and decompose $L^2(G)$.

Experimental results

Research questions

• RQ1 How can the Plancherel formula be extended to Whittaker functions on $p$-adic reductive groups?
• RQ2 What is the precise form of the Plancherel measure for Whittaker vectors in the $p$-adic setting?
• RQ3 How do $L$-functions and intertwining operators contribute to the spectral decomposition of $L^2(G)$?
• RQ4 What is the role of the minimal parabolic subgroup and its Levi component in the construction of Whittaker functionals?
• RQ5 How does the Whittaker model relate to the unitary dual of $G$?

Key findings

• The Plancherel formula for Whittaker functions on $G$ is derived using the spectral decomposition of $L^2(G)$ via Whittaker vectors.
• The Plancherel measure is expressed in terms of the standard $L$-functions and the normalization of intertwining operators.
• The decomposition of $L^2(G)$ into irreducible unitary representations is realized through Whittaker functionals on induced representations.
• The non-degenerate Whittaker functional on $Ind_{P_0}^G(\rho)$ is shown to be unique up to scalar, ensuring the spectral resolution is well-defined.
• The formula establishes a duality between the Whittaker model and the unitary dual of $G$, generalizing classical results to the $p$-adic case.
• The result provides a complete characterization of the $L^2$-spectrum of $G$ in terms of Whittaker vectors and $L$-functions.