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Sascha Orlik

Publications and source records attributed to Sascha Orlik.

16 recordsLinked to original sources
arXiv2026

On some non-principal locally analytic representations induced by cuspidal Lie algebra representations

Let $G$ be a split reductive $p$-adic Lie group. This paper is the first in a series on the construction of locally analytic $G$-representations which do not lie in the principal series. Here we consider the case of the general linear group $G=GL_{n+1}$ and locally analytic representations which are induced by cuspidal modules of the Lie algebra. We prove that they are ind-admissible and satisfy the homological vanishing criterion in the definition of supercuspidality in the sense of Kohlhaase.

math.RT↗
arXiv2025

On some non-principal locally analytic representations induced by Whittaker modules

Let G be a connected split reductive p-adic Lie group. This paper can be seen as a continuation of [12] and is about the construction of locally analytic G-representations which do not lie in the principal series. Here we consider locally analytic representations which are induced by Whittaker modules of the attached Lie algebra. We prove that they are inadmissible and topologically irreducible in case the Whittaker module is simple. On the other hand, we show that the naive Jacquet functor of these representations vanishes for all parabolic subgroups. However, they do not satisfy the definition of supercuspidality in the sense of Kohlhaase.

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arXiv2021

Equivariant vector bundles on Drinfeld's halfspace over a finite field

Let $\mathcal{X} \subset \mathbb{P}_k^d$ be Drinfeld's halfspace over a finite field $k$ and let $\mathcal{E}$ be a homogeneous vector bundle on $\mathbb{P}_k^d$. The paper deals with two different descriptions of the space of global sections $H^0(\mathcal{X},\mathcal{E})$ as $GL_{d+1}(k)$-representation. This is an infinite dimensional modular representation. Here we follow the ideas of \cite{O2,OS} treating the $p$-adic case. As a replacement for the universal enveloping algebra we consider both the crystalline universal enveloping algebra and the ring of differential operators on the flag variety with respect to $\mathcal{E}.$

math.AG↗
arXiv2019

The pro-étale cohomology of Drinfeld's upper half space

We determine the geometric pro-étale cohomology of Drinfeld's upper half space ${\mathcal X}$ over a p-adic field. The strategy is different from the one given by Colmez, Dospinescu and Niziol. It uses the approach developed in a former paper of the author describing global sections of equivariant vector bundles on ${\mathcal X}$.

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arXiv2014

Category ${\mathcal O}$ and locally analytic representations

For a split reductive group $G$ over a finite extension $L$ of ${\mathbb Q}_p$, and a parabolic subgroup $P \subset G$ we introduce a category ${\mathcal O}^P$ which is equipped with a forgetful functor to the parabolic category ${\mathcal O}^{\mathfrak p}$ of Bernstein, Gelfand and Gelfand. There is a canonical fully faithful embedding of a subcategory ${\mathcal O}^{\mathfrak p}_{\rm alg}$ of ${\mathcal O}^{\mathfrak p}$ into ${\mathcal O}^P$, which 'splits' the forgetful map. We then introduce functors from the category ${\mathcal O}^P$ to the category of locally analytic representations, thereby generalizing the authors' previous work where these functors had been defined on the category ${\mathcal O}^{\mathfrak p}_{\rm alg}$. It is shown that these functors are exact, and a criterion for the irreducibility of a representation in the image of this functor is proved.

math.RT↗
arXiv2014

The de Rham cohomology of Drinfeld's half space

Let X be Drinfeld's half space over a p-adic field K. The de Rham cohomology of X was first computed by Schneider and Stuhler. Afterwards there were given different proofs by Alon, de Shalit, Iovita and Spiess. This paper presents yet another approach for the determination of these invariants by analysing the de Rham complex of X from the viewpoint of recent results by the author.

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arXiv2014

The Jordan-Hölder series of the locally analytic Steinberg representation

We determine the composition factors of a Jordan-Hölder series including multiplicities of the locally analytic Steinberg representation. For this purpose we prove the acyclicity of the evaluated locally analytic Tits complex giving rise to the Steinberg representation. Further we describe some analogue of the Jacquet functor applied to the irreducible principal series representation constructed by Orlik and Strauch.

math.RT↗
arXiv2007

Equivariant vector bundles on Drinfeld's upper half space

Let X be Drinfeld's upper half space of dimension d over a finite extension K of Q_p. We construct for every homogeneous vector bundle F on the projective space P^d a GL_{d+1}(K)-equivariant filtration by closed K-Frechet spaces on F(X). This gives rise by duality to a filtration by locally analytic GL_{d+1}(K)-representations on the strong dual. The graded pieces of this filtration are locally analytic induced representations from locally algebraic ones with respect to maximal parabolic subgroups. This paper generalizes the cases of the canonical bundle due to Schneider and Teitelbaum and that of the structure sheaf by Pohlkamp.

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arXiv2005

The continuous cohomology of period domains over local fields

In this paper we consider period domains over local fields for quasi-split reductive groups. We compute the continuous l-adic cohomology with compact support of them in the case of a basic isocrystal. This paper is a continuation of [O2] where we considered the etale cohomology with torsion coefficients of these spaces.

math.NT↗
arXiv2004

On Extensions of generalized Steinberg Representations

Let F be a local non-archimedean field and let G be the group of F-valued points of a reductive algebraic group over F. In this paper we compute the Ext-groups of generalized Steinberg representations in the category of smooth G-representations with coefficients in a certain self-injective ring.

math.RT↗
arXiv1999

Kohomologie von Periodenbereichen ueber endlichen Koerpern

Periodenbereiche sind gewisse offene Unterraeume von verallgemeinerten Flaggenvarietaeten, welche durch Semistabilitaetsbedingungen beschrieben werden. In dem Fall eines endlichen Grundkoerpers bilden diese eine Zariski-offene Untervarietaet, im Fall eines lokalen Koerpers einen zulaessigen offenen Unterraum im Sinne der rigiden algebraischen Geometrie. In dieser Arbeit berechnen wir fuer den Fall eines endlichen Grundkoerpers die l-adische Kohomologie mit kompaktem Traeger dieser Periodenbereiche. Das Ergebnis bestaetigt eine Vermutung von Kottwitz und Rapoport.

math.AG↗