[Paper Review] Whittaker functions associated to newforms for GL(n) over p-adic fields

Michitaka Miyauchi|arXiv (Cornell University)|Jan 17, 2012

Advanced Algebra and Geometry5 references3 citations

TL;DR

This paper provides an explicit formula for Whittaker functions associated to newforms of irreducible generic representations of $\mathrm{GL}_n(F)$ over a $p$-adic field $F$, expressed in terms of Hecke eigenvalues and the $L$-factor of the representation. The key result shows that integrating such Whittaker functions over $\mathrm{GL}_1(F)$ yields the standard $L$-function of the representation, extending Shintani's formula for unramified cases to the newform setting via Hecke algebra actions and symmetric functions.

ABSTRACT

Let F be a non-archimedean local field of characteristic zero. Jacquet, Piatetski-Shapiro and Shalika introduced the notion of newforms for irreducible generic representations of GL_n(F). In this paper, we give an explicit formula for Whittaker functions associated to newforms on the diagonal matrices in GL_n(F).

Motivation & Objective

• To extend Shintani’s explicit formula for spherical Whittaker functions to the case of newforms for $\mathrm{GL}_n(F)$ over $p$-adic fields.
• To express Whittaker functions associated to newforms on diagonal matrices in terms of Hecke eigenvalues and the $L$-factor of the representation.
• To establish a zeta integral representation of the $L$-function using newform Whittaker functions.
• To provide a constructive formula for newform Whittaker functions on $BK_{c(\pi)}$, which is sufficient for computing Rankin-Selberg type zeta integrals.

Proposed method

• Use the Hecke algebra associated to the compact open subgroup $K_{c(\pi)}$ acting on the one-dimensional space of newforms, with action given by scalar multiplication via Hecke eigenvalues $\lambda_1, \dots, \lambda_{n-1}$.
• Define a normalized Whittaker function $\widetilde{w}(f) = q^{\sum (n-1-j)f_j} W(\varpi^f)$ to transform the functional equation into a system of difference equations.
• Solve the system of difference equations using elementary symmetric polynomials in variables $\mu_i$, related to the Langlands parameters $\alpha_i$ via $\mu_i = q^{(n-1)/2 - 1} \alpha_i$.
• Express the Whittaker function on diagonal matrices $\varpi^f = \mathrm{diag}(\varpi^{f_1}, \dots, \varpi^{f_{n-1}}, 1)$ as $W(\varpi^f) = \delta_B^{1/2}(\varpi^f) s_f(\alpha) W(1)$ when $f_1 \geq \dots \geq f_{n-1} \geq 0$, and zero otherwise.
• Use the Kirillov model and Iwasawa decomposition to ensure non-vanishing at some $\varpi^f$, and prove $W(1) \neq 0$ for non-zero newforms.
• Integrate the Whittaker function over $\mathrm{GL}_1(F) \subset \mathrm{GL}_n(F)$ via $Z(s,W) = \int_{F^\times} W(t(a)) |a|^{s - (n-1)/2} d^\times a$, showing it equals $L(s,\pi)$ when $W(1) = 1$.

Experimental results

Research questions

• RQ1 How can Shintani’s explicit formula for spherical Whittaker functions be generalized to newforms in $\mathrm{GL}_n(F)$ over $p$-adic fields?
• RQ2 What is the precise expression for Whittaker functions associated to newforms on diagonal matrices in $\mathrm{GL}_n(F)$?
• RQ3 How are the Hecke eigenvalues of the newform’s Hecke algebra related to the $L$-factor of the representation?
• RQ4 Can the $L$-factor of an irreducible generic representation $\pi$ be recovered from a zeta integral of its newform Whittaker function?
• RQ5 To what extent does the formula for Whittaker functions on $BK_{c(\pi)}$ suffice for computing zeta integrals in Rankin-Selberg theory?

Key findings

• The Whittaker function $W$ associated to a newform for $\mathrm{GL}_n(F)$ satisfies $W(\varpi^f) = \delta_B^{1/2}(\varpi^f) s_f(\alpha) W(1)$ for $f_1 \geq \dots \geq f_{n-1} \geq 0$, and vanishes otherwise, where $s_f(\alpha)$ is the complete homogeneous symmetric polynomial in the Langlands parameters $\alpha_1, \dots, \alpha_n$.
• The Hecke eigenvalues $\lambda_i$ are related to the $L$-factor via $L(s,\pi) = \left( \sum_{i=0}^{n-1} (-1)^i \lambda_i q^{i(i-1)/2 - i((n-1)/2 + s)} \right)^{-1}$, and the parameters $\mu_i = q^{(n-1)/2 - 1} \alpha_i$ solve the difference equations for $\widetilde{w}(f)$.
• The zeta integral $Z(s,W) = \int_{F^\times} W(t(a)) |a|^{s - (n-1)/2} d^\times a$ equals the standard $L$-function $L(s,\pi)$ when $W$ is a newform with $W(1) = 1$.
• The formula determines the Whittaker function on $BK_{c(\pi)}$, which is sufficient for computing zeta integrals when all data are newforms.
• The space of newforms is one-dimensional, and $W(1) \neq 0$ for any non-zero newform, ensuring normalization is possible.
• The result generalizes Shintani’s formula for unramified representations to the ramified case via the conductor $c(\pi) > 0$ and Hecke algebra methods.