arXiv subjects

Paul Philippe

Publications and source records attributed to Paul Philippe.

4 recordsLinked to original sources

arXiv2026

Oort's conjecture for split unitary Shimura varieties

We prove that generically on the basic stratum of split unitary Shimura varieties, the universal abelian variety with endomorphism structure and polarization has automorphism group $\{\pm 1\}$, except for a few degenerate cases. This is a direct analogue of Oort's conjecture on automorphisms of supersingular abelian varieties. On the way, we explicitly compute the generic automorphism group of the universal $p$-divisible group in a given basic isogeny class.

math.AG↗

arXiv2024

On affine Kazhdan-Lusztig R-polynomials for Kac-Moody groups

In 2019, D. Muthiah proposed a strategy to define affine Kazhdan-Lusztig $R$-polynomials for Kac-Moody groups. Since then, Bardy-Panse, the first author and Rousseau have introduced the formalism of twin masures and the authors have extended combinatorial results from affine root systems to general Kac-Moody root systems in a previous article. In this paper, we use these results to explicitly define affine $R$-Kazhdan-Lusztig polynomials for Kac-Moody groups. The construction is based on a path model lifting to twin masures. Conjecturally, these polynomials count the cardinality of intersections of opposite affine Schubert cells, as in the case of reductive groups.

math.RT↗

arXiv2024

Quantum roots for Kac-Moody root systems and finiteness properties of the Kac-Moody affine Bruhat order

Let $G$ be a split Kac-Moody group over a local field. In their study of the Iwahori-Hecke algebra of $G$, A.Braverman, D. Kazhdan and M. Patnaik defined a partial order - called the affine Bruhat order - on the extended affine Weyl semi-group $W^+$ of $G$. In this paper, we study finiteness questions for covers and co-covers of $W^+$, generalizing results of A. Welch. In particular we prove that the intervals for this order are finite. Our results rely on the finiteness of the set of quantum roots of arbitrary Kac-Moody root systems, which we prove. We also obtain a classification of quantum roots.

math.RT↗

arXiv2023

Grading of affine Weyl semi-groups of Kac-Moody type

For any Kac-Moody root data $\mathcal D$, D. Muthiah and D. Orr have defined a partial order on the semi-direct product $W^+$ of the integral Tits cone with the vectorial Weyl group of $\mathcal D$, and a strictly compatible $\mathbb Z$-valued length function. We classify covers for this order and show that this length function defines a $\mathbb Z$-grading of $W^+$, generalizing the case of affine ADE root systems and giving a positive answer to a conjecture of Muthiah and Orr.

math.RT↗