Geometrization of principal series representations of reductive groups

Kamgarpour, Masoud; Schedler, Travis

Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 2273-2330.

En théorie des représentations, on cherche souvent à écrire des représentations réalisées dans des espaces de fonctions invariantes comme les fonctions trace de faisceaux pervers équivariants. Dans le cas des représentations de la série principale d’un groupe réductif $G$ connexe scindé sur un corps local, il existe une description des familles de telles représentations realisées dans des espaces de fonctions sur $G$ invariantes sous l’action de translation du sous-groupe d’Iwahori ou d’un sous-groupe compact ouvert plus petit approprié, comme l’ont etudié Howe, Bushnell et Kutzko, Roche, et d’autres. Dans cet article, nous construisons des catégories de faisceaux pervers dont les traces redonnent les families associées aux caractères réguliers de $T\left({𝔽}_q\left[\left[t\right]\right]\right)$, et démontrons des conjectures de Drinfeld pour leur structure. Nous proposons également des conjectures sur la géométrisation des familles associées à des caractères plus généraux.

In geometric representation theory, one often wishes to describe representations realized on spaces of invariant functions as trace functions of equivariant perverse sheaves. In the case of principal series representations of a connected split reductive group $G$ over a local field, there is a description of families of these representations realized on spaces of functions on $G$ invariant under the translation action of the Iwahori subgroup, or a suitable smaller compact open subgroup, studied by Howe, Bushnell and Kutzko, Roche, and others. In this paper, we construct categories of perverse sheaves whose traces recover the families associated to regular characters of $T\left({𝔽}_q\left[\left[t\right]\right]\right)$, and prove conjectures of Drinfeld on their structure. We also propose conjectures on the geometrization of families associated to more general characters.

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