[Paper Review] Transfer of R-groups between p-adic inner forms of SL_n

Kwangho Choiy, David Theo Goldberg|arXiv (Cornell University)|Nov 21, 2012

Advanced Algebra and Geometry17 references4 citations

TL;DR

This paper establishes the transfer of Knapp-Stein R-groups from inner forms of $SL_n(F)$ to the split group $SL_n(F)$, where $F$ is a $p$-adic field. Using the generalized Jacquet-Langlands correspondence and Plancherel measure invariance, it proves that the R-group for an inner form embeds into the R-group of the split form, with the quotient isomorphic to a subgroup of the character group of the derived group, and confirms the R-group is isomorphic to Arthur’s endoscopic R-group as predicted by Arthur.

ABSTRACT

We study the Knapp-Stein $R$--groups for inner forms of the split group $SL_n(F),$ with $F$ a $p$--adic field of characteristic zero. Thus, we consider the groups $SL_m(D),$ with $D$ a central division algebra over $F$ of dimension $d^2,$ and $m=n/d.$ We use the generalized Jacquet-Langlands correspondence and results of the first named author to describe the zeros of Plancherel measures. Combined with a study of the behavior of the stabilizer of representations by elements of the Weyl group we are able to determine the Knapp-Stein $R$--groups in terms of those for $SL_n(F).$ We show the $R$--group for the inner form embeds as a subgroup of the $R$--group for the split form, and we characterize the quotient. We are further able to show the Knapp-Stein $R$--group is isomorphic to the Arthur, or Endoscopic $R$--group as predicted by Arthur. Finally, we give some results on multiplicities and actions of Weyl groups on $L$--packets.

Motivation & Objective

• To understand the structure of Knapp-Stein R-groups for non-split inner forms of $SL_n(F)$, where $F$ is a $p$-adic field of characteristic zero.
• To determine how R-groups for inner forms $SL_m(D)$ (with $D$ a central division algebra over $F$) relate to those for the split group $SL_n(F)$.
• To verify that the Knapp-Stein R-group for inner forms is isomorphic to Arthur’s endoscopic R-group, as predicted by Arthur’s conjectures.
• To analyze the action of the Weyl group on $L$-packets and determine multiplicities in the tempered spectrum of inner forms.

Proposed method

• Utilize the generalized Jacquet-Langlands correspondence between $L$-packets of Levi subgroups of $GL_n(F)$ and $GL_m(D)$ to relate representations on inner forms.
• Apply results from the first author on Plancherel measure zeros to show that the set of roots corresponding to zero Plancherel measures is preserved under the correspondence.
• Use the Weyl group action on representations to characterize the stabilizer $W(\sigma)$ of a representation $\sigma$ in the $L$-packet.
• Establish that $W(\sigma^\prime) \subset \bar{W}$, where $\bar{W}$ is the set of Weyl group elements preserving $\widetilde{\sigma}^\prime$ up to twist by a character of $\widetilde{M}^\prime/M^\prime$, and relate this to the R-group structure.
• Prove that the R-group for the inner form $SL_m(D)$ embeds into the R-group for $SL_n(F)$, with the quotient isomorphic to a subgroup of $X_{M^\prime}(\phi)/X(\sigma^\prime)$.
• Use the fact that $\mu_\beta(\sigma) = \mu_\beta(\sigma^\prime)$ for all roots $\beta$, implying $W^\prime_\sigma = W^\prime_{\sigma^\prime}$, to reduce the problem to comparing stabilizers.

Experimental results

Research questions

• RQ1 How do the Knapp-Stein R-groups of inner forms $SL_m(D)$ of $SL_n(F)$ relate to those of the split group $SL_n(F)$?
• RQ2 Does the generalized Jacquet-Langlands correspondence preserve the Plancherel measure zeros that define the R-group?
• RQ3 Is the Knapp-Stein R-group for an inner form isomorphic to Arthur’s endoscopic R-group, as predicted by Arthur’s conjectures?
• RQ4 What is the structure of the Weyl group action on $L$-packets for inner forms, and how does it affect multiplicities?
• RQ5 How does the stabilizer of a representation in the Weyl group relate to the character twists of its lift under the Jacquet-Langlands correspondence?

Key findings

• The R-group for an inner form $SL_m(D)$ embeds as a subgroup into the R-group for the split group $SL_n(F)$, with the quotient isomorphic to a subgroup of $X_{M^\prime}(\phi)/X(\sigma^\prime)$.
• The stabilizer $W(\sigma^\prime)$ of a representation $\sigma^\prime$ on the inner form corresponds to the set of Weyl group elements $w$ such that ${}^w\widetilde{\sigma}^\prime \simeq \widetilde{\sigma}^\prime \otimes \eta$ for some character $\eta \in (\widetilde{M}^\prime/M^\prime)^D$, establishing a precise link to the R-group structure.
• The Plancherel measure zeros are preserved under the generalized Jacquet-Langlands correspondence, so $W^\prime_\sigma = W^\prime_{\sigma^\prime}$, which reduces the R-group comparison to comparing $W(\sigma)$ and $W(\sigma^\prime)$.
• The Knapp-Stein R-group for the inner form is isomorphic to Arthur’s endoscopic R-group, confirming Arthur’s prediction in this setting.
• When the $L$-packet $\Pi_\phi(M^\prime)$ is a singleton, the R-groups for corresponding representations on the split and inner forms are isomorphic: $R_\sigma = R_{\sigma^\prime}$.
• The action of the Weyl group on $L$-packets is controlled by the character twists of the Langlands parameters, and the structure of the $R$-group reflects this action through the quotient $\bar{W}/W(\sigma^\prime)$.