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Geometric stabilisation via \(p\)-adic integration. (English) Zbl 1465.11229

Langlands and Shelstad’s fundamental lemma relates the \(\kappa\)-orbital integrals of a reductive group \(G\) to the stable orbital integrals of its \(\kappa\)-endoscopic group. To prove the lemma, building on the work of many others, Ngô reformulated it using the geometry of the Hitchin fibration, a certain map \[ \widetilde{\mathcal M}_G \to \widetilde{\mathsf A}_G \] from a stack to a variety [B. C. Ngô, Publ. Math., Inst. Hautes Étud. Sci. 111, 1–169 (2010; Zbl 1200.22011)]. The Hitchin fibration depends on a quasi-split reductive group scheme \(G\) over a curve \(X\), itself defined over a finite field \(k\) of cardinality \(q\). Ngô’s reformulation, known as geometric stabilisation, concerns the sizes of the fibers of the Hitchin fibration.
More precisely, for each \(a\in\widetilde{\mathsf A}_G\), there is a certain group \(\pi_0(\widetilde{\mathcal P}_{G,a})\) that acts on the compactly-supported cohomology \(H^*_c(\widetilde{\mathcal M}_{G,a},\overline{\mathbb Q}_\ell)\). For each character \(\kappa\) of \(\pi_0(\widetilde{\mathcal P}_{G,a})\), one defines the \(\kappa\)-point count of the fiber \(\widetilde{\mathcal M}_{G,a}\) as the graded trace of Frobenius acting on the \(\kappa\)-isotypical component of the cohomology. Of particular importance is the stable point count, the \(\kappa\)-point count for \(\kappa = 1\).
Let \(H\) be an endoscopic group of \(G\). There is a canonical closed immersion \(\widetilde{\mathsf A}_H\to\widetilde{\mathsf A}_G\), and for each \(a\in\widetilde{\mathsf A}_H\), a certain character \(\kappa_a\). Geometric stabilisation asserts that up to a scaling factor depending only on \(q\), \(G\), and \(H\), the \(\kappa_a\)-point count of the fiber for \(G\) equals the stable point count of the fiber for \(H\), both fibers being taken over \(a\).
The article under review gives a new proof of Ngô’s geometric stabilisation lemma, building on earlier work of the authors in the case of the special linear group [M. Groechenig et al., Invent. Math. 221, No. 2, 505–596 (2020; Zbl 1451.14123)]. Unlike Ngô’s original proof, which relied on the decomposition theorem and codimension estimates, here the key tool is \(p\)-adic integration.
A classical construction of Weil defines a canonical measure on a smooth scheme \(X\) over the ring of integers \(\mathcal O_F\) of a nonarchimedean local field \(F\) with residue field \(k\). Weil’s measure has the property that up to a scaling factor depending only on \(q\) and the relative dimension of \(X\), the measure of \(X(\mathcal O_F)\) is the cardinality of \(X(k)\).
To prove geometric stabilisation, the authors generalize Weil’s measure to one defined on the relevant space in this setting, a certain \(F\)-analytic manifold \(M(\mathcal O_F)^\natural\). This manifold admits a specialization map to a twisted inertia stack. As it was for Weil, the volume of \(M(\mathcal O_F)^\natural\) is proportional to the cardinality of this inertia stack. In this generality the constant of proportionality depends on \(q\) and a certain weight function, and the cardinality of the inertia stack takes isotropy groups into account.
This generalization of Weil’s result is only one of many tools used in this article to prove geometric stabilisation. The article is written clearly, and its authors are to be commended for their deep understanding of the many areas of mathematics that inform their proof.

MSC:

11S37 Langlands-Weil conjectures, nonabelian class field theory
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
14H60 Vector bundles on curves and their moduli
20G40 Linear algebraic groups over finite fields
14D24 Geometric Langlands program (algebro-geometric aspects)
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References:

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