[Paper Review] Distinguished representations for SL(n)

U. K. Anandavardhanan, Dipendra Prasad|arXiv (Cornell University)|Dec 4, 2016

Advanced Algebra and Geometry12 references4 citations

TL;DR

This paper establishes a precise formula for the dimension of the space of $\mathrm{SL}_n(F)$-invariant linear functionals on irreducible admissible generic representations $\pi$ of $\mathrm{SL}_n(E)$, where $E/F$ is a quadratic extension of local or finite fields. It proves that this dimension equals the size of the fiber of the base change map from $\mathrm{SU}_n(F)$ to $\mathrm{SL}_n(E)$, confirming a conjecture of Prasad for $G = \mathrm{SL}_n$. The result generalizes earlier work on $n=2$ and odd $n$, and introduces a Whittaker model condition as a key criterion for distinction.

ABSTRACT

For $E/F$ a quadratic extension of local fields, and $π$ an irreducible admissible generic representation of $SL_n(E)$, we calculate the dimension of $Hom_{SL_n(F)}[π,C]$ and relate it to fibers of the base change map corresponding to base change of representations of $SU_n(F)$ to $SL_n(E)$ as suggested in a recent work of the second author. We also deal with finite fields.

Motivation & Objective

• To compute $\dim_{\mathbb{C}} \mathrm{Hom}_{\mathrm{SL}_n(F)}[\pi, \mathbb{C}]$ for irreducible admissible generic representations $\pi$ of $\mathrm{SL}_n(E)$, where $E/F$ is a quadratic extension.
• To relate this dimension to the fibers of the base change map from $\mathrm{SU}_n(F)$ to $\mathrm{SL}_n(E)$, as suggested in [Pra16].
• To extend previous results on distinction for $n=2$ and odd $n$ to the general case, particularly for even $n$.
• To establish a characterization of $\mathrm{SL}_n(F)$-distinguished representations via their Whittaker models over $N(E)/N(F)$.

Proposed method

• Uses the base change map $P\Phi: H^1(W_F', \mathrm{PGL}_n(\mathbb{C})[\tau]) \to H^1(W_E', \mathrm{PGL}_n(\mathbb{C}))$ to relate representations of $\mathrm{SU}_n(F)$ to those of $\mathrm{SL}_n(E)$.
• Applies results from Matringe on $\mathrm{GL}_n(F)$-distinction of generic representations of $\mathrm{GL}_n(E)$ to analyze $\mathrm{SL}_n(F)$-invariant functionals.
• Relies on the fact that $\mathrm{PGL}_n(F)$ acts transitively on $L$-packets of $\mathrm{SL}_n(E)$, ensuring that the dimension of $\mathrm{Hom}_{\mathrm{SL}_n(F)}[\pi, \mathbb{C}]$ is constant on $\mathrm{PGL}_n(F)$-conjugacy classes.
• Uses the identification of $\mathrm{SL}_n(E)$-parameters with $\mathrm{GL}_n(E)$-parameters up to twisting by characters, and relates $\mathrm{SU}_n(F)$-parameters to $\mathrm{GL}_n(E)$-parameters via the base change map $\Phi$.
• Employs the notion of strong and weak equivalence of Langlands parameters to compute the size of fibers of the base change map.
• Establishes that $\mathrm{SL}_n(F)$-distinguished generic representations must admit a Whittaker model for a non-degenerate character of $N(E)/N(F)$, using results from [AM17] on tempered and unitary representations.

Experimental results

Research questions

• RQ1 What is the precise dimension of $\mathrm{Hom}_{\mathrm{SL}_n(F)}[\pi, \mathbb{C}]$ for an irreducible admissible generic representation $\pi$ of $\mathrm{SL}_n(E)$?
• RQ2 How does this dimension relate to the fibers of the base change map from $\mathrm{SU}_n(F)$ to $\mathrm{SL}_n(E)$?
• RQ3 Under what conditions does a generic representation $\pi$ of $\mathrm{SL}_n(E)$ admit a non-zero $\mathrm{SL}_n(F)$-invariant linear functional?
• RQ4 Is the Whittaker model condition over $N(E)/N(F)$ necessary and sufficient for $\mathrm{SL}_n(F)$-distinction of generic representations?

Key findings

• The dimension of $\mathrm{Hom}_{\mathrm{SL}_n(F)}[\pi, \mathbb{C}]$ is equal to the size of the fiber of the base change map $P\Phi$ over the Langlands parameter $\rho_\pi$ of $\pi$, i.e., $\dim_{\mathbb{C}} \mathrm{Hom}_{\mathrm{SL}_n(F)}[\pi, \mathbb{C}] = |P\Phi^{-1}(\rho_\pi)|$.
• A generic representation $\pi$ of $\mathrm{SL}_n(E)$ is $\mathrm{SL}_n(F)$-distinguished if and only if its Langlands parameter $\rho_\pi$ lies in the image of the base change map $P\Phi$.
• The $\mathrm{SL}_n(F)$-distinction of $\pi$ is equivalent to $\pi$ admitting a Whittaker model for a non-degenerate character of $N(E)/N(F)$, where $N$ is the unipotent radical of a Borel subgroup.
• For $n$ even, the symmetric space $({\rm SL}_n(E), {\rm SL}_n(F))$ is not a Gelfand pair, and multiplicity can exceed one, in contrast to the odd $n$ case.
• The result confirms the general conjecture of Prasad [Pra16] for $G = \mathrm{SL}_n$, relating the dimension of invariant functionals to fibers of the base change map.
• The formula holds uniformly for both $p$-adic and finite fields, extending previous results that were limited to $n=2$ or odd $n$.