[Paper Review] Remarks on the Fundamental Lemma for stable twisted Endoscopy of Classical Groups

Joachim Ballmann, Rainer Weissauer+1 more|arXiv (Cornell University)|Jan 31, 2013

Advanced Algebra and Geometry21 references3 citations

TL;DR

This paper establishes the fundamental lemma for stable twisted endoscopy in classical groups by reducing the $Θ$-twisted orbital integral matching problem to a conjectural statement—called the $BC$-conjecture—on ordinary stable orbital integrals for $\mathrm{SO}_{2n+1}$ and $\mathrm{Sp}_{2n}$. Using the Kazhdan trick and topological Jordan decomposition, it proves that matching semisimple elements in $\widetilde{\mathrm{SO}}_{2n+2}$ and $\mathrm{Sp}_{2n}$ yield equal stable orbital integrals, assuming the $BC_n$-conjecture holds for smaller ranks.

ABSTRACT

The twisted fundamental lemmas for three series of stable twisted endoscopy (Sp_{2n} to PGL_{2n+1}, GSpin_{2n+1} to GL_{2n} imes GL_1, Sp_{2n} to SO_{2n+2}) will be reduced to a fundamental-lemma-like statement for ordinary (i.e. untwisted) stable orbital integrals on the groups $\SO_{2n+1}$ of type $B_n$ resp. $\SP_{2n}$ of type $C_n$. This manusript, written around 2001, anticipates similar results of Waldspurger (2008), and may be used to reduce the twisted fundamental lemma to the work of Ngo. The original aim was to reduce the fundumantal lemma for the lift from Sp_4 to PGL_5 to the twisted fundamental lemma for the lift from GSp_4=GSpin_5 to Gl_4 imes GL_1.

Motivation & Objective

• To establish the fundamental lemma for stable twisted endoscopy in classical groups, particularly for the lift from $\mathrm{Sp}_{2n}$ to $\widetilde{\mathrm{PGL}}_{2n+1}$ and related groups.
• To reduce the $\Theta$-twisted fundamental lemma to a conjectural statement on ordinary stable orbital integrals for $\mathrm{SO}_{2n+1}$ and $\mathrm{Sp}_{2n}$, termed the $BC_n$-conjecture.
• To prove that matching semisimple elements in $\widetilde{\mathrm{SO}}_{2n+2}(F)$ and $\mathrm{Sp}_{2n}(F)$ yield equal stable orbital integrals under the assumption of the $BC_m$-conjecture for $m \leq n$.
• To provide a foundational tool for the theory of automorphic representations and Galois representations on Shimura varieties associated with $\mathrm{GSp}_4$.
• To clarify the relationship between twisted and untwisted orbital integrals via topological Jordan decomposition and centralizer analysis in $p$-adic groups.

Proposed method

• Applies the Kazhdan trick (Lemma 5.5) to relate $\Theta$-twisted orbital integrals on $\widetilde{G}$ to stable orbital integrals on the endoscopic group $H$.
• Uses topological Jordan decomposition to split elements in $\widetilde{\mathrm{SO}}_{2n+2}(F)$ and $\mathrm{Sp}_{2n}(F)$ into semisimple and unipotent parts, enabling reduction to matching components.
• Employs $BC$-matching between regular topologically unipotent elements in $\mathrm{SO}_{2n+1}(\mathcal{O}_F)$ and $\mathrm{Sp}_{2n}(\mathcal{O}_F)$ to equate their stable orbital integrals.
• Analyzes conjugacy classes in $\widetilde{\mathrm{SO}}_{2n+2}(F)$ via representatives in $\mathrm{O}_{2n+2}(\mathcal{O}_F)$, distinguishing between $\det = 1$ and $\det = -1$ components.
• Applies Lemma 5.7 to decompose stable orbital integrals over products of centralizers, using the fact that $\mathrm{SO}_{2n+2}^s$ and $\mathrm{SO}_{2n+2}^{s''}$ are isomorphic as algebraic groups.
• Uses the $\Theta$-action to define $\widetilde{G} = G \rtimes \langle \Theta \rangle$, with $\Theta(g) = J \cdot {}^t g^{-1} \cdot J^{-1}$, to model the twisted endoscopy.

Experimental results

Research questions

• RQ1 Under what conditions do $\Theta$-twisted orbital integrals on $\widetilde{\mathrm{PGL}}_{2n+1}$ match stable orbital integrals on $\mathrm{Sp}_{2n}$?
• RQ2 How can the fundamental lemma for twisted endoscopy be reduced to a statement about ordinary stable orbital integrals on $\mathrm{SO}_{2n+1}$ and $\mathrm{Sp}_{2n}$?
• RQ3 What role does $BC$-matching play in relating stable orbital integrals across classical groups with outer automorphisms?
• RQ4 How do topological Jordan decompositions and centralizer structures affect the matching of orbital integrals in $p$-adic groups?
• RQ5 What is the precise relationship between the stable conjugacy classes in $\mathrm{Sp}_{2n}(F)$ and $\widetilde{\mathrm{SO}}_{2n+2}(F)$ under the given matching conditions?

Key findings

• The fundamental lemma for the stable lift from $\mathrm{Sp}_4$ to $\widetilde{\mathrm{PGL}}_5$ is established under the assumption that the $BC_1$-conjecture holds.
• For $\gamma \in \mathrm{Sp}_{2n}(F)$ and $g \in \widetilde{\mathrm{SO}}_{2n+2}(F)$ matching semisimple elements, the stable orbital integrals satisfy $O^{st}_{g}(1,\widetilde{\mathrm{SO}}_{2n+2}) = O^{st}_{\gamma}(1,\mathrm{Sp}_{2n})$ if the $BC_n$-conjecture holds for all $m \leq n$.
• If $\gamma \in \mathrm{Sp}_{2n}(F)$ does not match any element in $\widetilde{\mathrm{SO}}_{2n+2}(F)$, then $O^{st}_{\gamma}(1,\mathrm{Sp}_{2n}) = 0$, confirming the matching condition is necessary.
• The proof relies on decomposing elements into topological Jordan components: $u_+, u_-, u_*$ for $\mathrm{SO}_{2n+1}$ and $v_+, v_-, v_*$ for $\mathrm{Sp}_{2n}$, with $BC$-matching between $u_\pm$ and $v_\pm$.
• The centralizer $\mathrm{SO}_{2n+2}^s$ for $s \in \mathrm{O}_{2n+2}(\mathcal{O}_F)$ of determinant $-1$ is isomorphic to a product of $\mathrm{SO}_{2r_{\pm}+1}$ and a centralizer in $\mathrm{Sp}_{2g}$, enabling integral decomposition.
• The factor of $\frac{1}{2}$ from the two connected components of the centralizer is canceled by the factor of $2$ from the two conjugacy classes of $s$ and $s''$, preserving the equality in the orbital integral sum.