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Peter Fiebig

Publications and source records attributed to Peter Fiebig.

At least 19 recordsLinked to original sources
arXiv2023

Representations and binomial coefficients

For a root system R, a field K and a "choice of coefficients in K" we define a category of graded spaces with operators and study some of its properties. Then we assume that the coefficients are given by quantum binomials. We use basic arithmetic properties of binomial coefficients (such as q-versions of Lucas' theorem and the Pfaff-Saalschütz identity) to construct a Frobenius pull back functor and prove a Steinberg's tensor product formula. Then we show that these results imply the corresponding results for reductive algebraic groups in positive characteristics and for quantum groups at roots of unity.

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arXiv2022

Quantum tilting modules over local rings

We show that tilting modules for quantum groups over local Noetherian domains exist and that the indecomposable tilting modules are parametrized by their highest weight. For this we introduce a model category ${\mathcal X}={\mathcal X}_{\mathscr A}(R)$ associated with a Noetherian ${\mathbb Z}[v,v^{-1}]$-domain ${\mathscr A}$ and a root system $R$. We show that if ${\mathscr A}$ is of quantum characteristic $0$, the model category contains all $U_{\mathscr A}$-modules that admit a Weyl filtration. If ${\mathscr A}$ is in addition local, we study torsion phenomena in the model category. This leads to a construction of torsion free objects in ${\mathcal X}$. We show that these correspond to tilting modules for the quantum group associated with ${\mathscr A}$ and $R$.

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arXiv2022

Periodicity for subquotients of the modular category $\mathcal{O}$

In this paper we study the category $\mathcal{O}$ over the hyperalgebra of a reductive algebraic group in positive characteristics. For any locally closed subset $\mathcal{K}$ of weights we define a subquotient $\mathcal{O}_{[\mathcal{K}]}$ of $\mathcal{O}$. It has the property that its simple objects are parametrized by elements in $\mathcal{K}$. We then show that $\mathcal{O}_{[\mathcal{K}]}$ is equivalent to $\mathcal{O}_{[\mathcal{K}+p^lγ]}$ for any dominant weight $γ$ if $l>0$ is an integer such that $\mathcal{K}\cap (\mathcal{K}+p^lη)=\emptyset$ for all dominant weights $η$. This allows one, for example, to restrict attention to subquotients inside the dominant (or the antidominant) chamber.

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arXiv2021

Tilting modules and torsion phenomena

We construct families of representations for quantum groups over $\mathbb{Z}[v,v^{-1}]$-algebras that interpolate between Weyl modules and tilting modules. These families might be candidates for objects with characters satisfying the {\em generations of characters} philosophy of Lusztig and Lusztig-Williamson.

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arXiv2021

Periodicity of irreducible modular and quantum characters

For a root system R, a field K and an invertible element q in K let U be the associated quantum group, defined via Lusztig's divided powers construction. We study the irreducible characters of this algebra with integral (but not necessarily dominant) highest weight. If the l-th cyclotomic polynomial vanishes when evaluated at q, then these characters exhibit a certain l-periodicity.

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arXiv2020

Lefschetz operators, Hodge-Riemann forms, and representations

For a field of characteristic $\ne 2$ we study vector spaces that are graded by the weight lattice of a root system, and are endowed with linear operators in each simple root direction. We show that these data extend to a graded semisimple representation of the corresponding Lie algebra if and only if there exists a bilinear form that satisfies properties (roughly) analogous to those of the Hodge-Riemann forms in complex geometry. In the second part of the article we replace the field by the $p$-adic integers (with $p\ne 2$) and show that in this case the existence of a certain bilinear form is equivalent to the existence of a structure of a tilting module for the associated simply connected $p$-adic Chevalley group.

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arXiv2020

Sheaves on the alcoves and modular representations II

We relate the category of sheaves on alcoves that was constructed in "Sheaves on the alcoves and modular representations I" to the representation theory of reductive algebraic groups. In particular, we show that its indecomposable projective objects encode the simple rational characters of a reductive algebraic group in all characteristics above the Coxeter number.

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arXiv2020

Sheaves on the alcoves and modular representations I

We consider the set of affine alcoves associated with a root system R as a topological space and consider a certain category S of sheaves of Z-modules on this space. Here Z is the structure algebra of the root system over a field k. To any wall reflection we associate a wall crossing functor on S. In the companion article "Sheaves on the alcoves and modular representations II" we prove that S encodes the simple rational characters of the connected, simply connected algebraic group with root system R over k, in the case that k is algebraically closed with characteristic above the Coxeter number.

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arXiv2017

Sheaves on the alcoves I: Projectivity and wall crossing functors

This paper is the first in a series of papers in which we define and study a category of "sheaves of $\mathcal Z$-modules on the set of alcoves" that carries important information on the category of representations of semisimple Lie algebras in positive characteristics. Here, we define this category and study the structure of its projective objects using a new version of wall crossing functors.

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arXiv2015

Filtered moment graph sheaves

We introduce the notion of (co-)filtered sheaves on quotients of moment graphs by a group action. We then introduce a (co-)filtered version of the canonical sheaves of Braden and MacPherson and show that their global sections are the indecomposable projective objects in a suitably defined exact category.

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arXiv2013

Moment graphs in representation theory and geometry

This paper reviews the moment graph technique that allows to translate certain representation theoretic problems into geometric ones. For simplicity we restrict ourselves to the case of semisimple complex Lie algebras. In particular, we show how the original Kazhdan-Lusztig conjecture on the characters of irreducible highest weight representations can be translated into a multiplicity problem for parity sheaves on the (Langlands dual) flag variety.

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arXiv2011

Parity sheaves, moment graphs and the p-smooth locus of Schubert varieties

We show that, with coefficients in a field or a complete local ring k, the Braden-MacPherson algorithm computes the stalks of parity sheaves with coefficients in k. As a consequence we deduce that the Braden-MacPherson algorithm may be used to calculate the characters of tilting modules for algebraic groups and show that the p-smooth locus of (Kac-Moody) Schubert varieties agrees with the rationally smooth locus, if the underlying Bruhat graph satisfies a GKM-condition.

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arXiv2011

The linkage principle for restricted critical level representations of affine Kac-Moody algebras

We study the restricted category O for an affine Kac--Moody algebra at the critical level. In particular, we prove the first part of the Feigin-Frenkel conjecture: the linkage principle for restricted Verma modules. Moreover, we prove a version of the BGGH-reciprocity principle and we determine the block decomposition of the restricted category O. For the proofs we need a deformed version of the classical structures, so we mostly work in a relative setting.

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arXiv2010

On the restricted Verma modules at the critical level

We study the restricted Verma modules of an affine Kac-Moody algebra at the critical level with special emphasis on their Jordan-H"older multiplicities. The Feigin-Frenkel conjecture gives a formula for these multiplicities that involves the periodic Kazhdan-Lusztig polynomials. We prove this conjecture for all subgeneric blocks and for the case of anti-dominant simple subquotients.

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