Local integrability results in harmonic analysis on reductive groups in large positive characteristic

Cluckers, Raf; Gordon, Julia; Halupczok, Immanuel

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 6, pp. 1163-1195.

Soit $𝐆$ un groupe algébrique réductif connexe au-dessus d'un corps local non archimédien $𝕂$, et soit $𝔤$ son algèbre de Lie. D'après un théorème de Harish-Chandra, si $𝕂$ est de caractéristique zéro, alors les transformés de Fourier d'intégrales orbitales sont représentés, sur l'ensemble des éléments réguliers de $𝔤\left(𝕂\right)$, par des fonctions localement constantes, qui, si on les étend par zéro à tout $𝔤\left(𝕂\right)$, sont localement intégrables. Dans ce papier, nous démontrons que ces fonctions sont en fait des spécialisations de fonctions motiviques constructibles exponentielles. En combinant ceci avec le principe de transfert d'intégrabilité de [8], nous obtenons que le théorème de Harish-Chandra est valable aussi quand $𝕂$ est un corps local non archimédien de caractéristique positive suffisamment grande. Sous l'hypothèse que l'application exponentielle feinte existe, ceci implique aussi l'intégrabilité locale des caractères de Harish-Chandra de représentations admissibles de $𝐆\left(𝕂\right)$, où $𝕂$ est un corps d'équicaractéristique suffisamment grande (en fonction de la donnée radicielle de $𝐆$).

Let $𝐆$ be a connected reductive algebraic group over a non-Archimedean local field $𝕂$, and let $𝔤$ be its Lie algebra. By a theorem of Harish-Chandra, if $𝕂$ has characteristic zero, the Fourier transforms of orbital integrals are represented on the set of regular elements in $𝔤\left(𝕂\right)$ by locally constant functions, which, extended by zero to all of $𝔤\left(𝕂\right)$, are locally integrable. In this paper, we prove that these functions are in fact specializations of constructible motivic exponential functions. Combining this with the Transfer Principle for integrability of [8], we obtain that Harish-Chandra's theorem holds also when $𝕂$ is a non-Archimedean local field of sufficiently large positive characteristic. Under the hypothesis that mock exponential map exists, this also implies local integrability of Harish-Chandra characters of admissible representations of $𝐆\left(𝕂\right)$, where $𝕂$ is an equicharacteristic field of sufficiently large (depending on the root datum of $𝐆$) characteristic.

Classification : 43A80, 22E50; 14E18, 03C99
Keywords: Harish-Chandra characters, orbital integrals, Fourier transform, local integrability, reductive group over a local field.
Mot clés : Caractères de Harish-Chandra, intégrales orbitales, transformés de Fourier, intégrabilité locale, groupes réductifs au-dessus d'un corps local.

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