An upper bound on the exceptional characteristics for Lusztig's character formula - Fiebig, Peter (Journal für die reine und angewandte Mathematik, 2012)
We develop a Lefschetz theory in a combinatorial category associated to a root system and derive an upper bound on the exceptional characteristics for Lusztig's formula for the simple rational characters of a reductive algebraic group. Our bound is huge compared to the Coxeter number. © 2012 Walter de Gruyter Berlin · Boston.

Centers and translation functors for category O over symmetrizable Kac-Moody algebras - Fiebig, Peter (Mathematische Zeitschrift, 2003)
We show that the structure of blocks outside the critical hyperplanes of category O over any symmetrizable Kac-Moody algebra depends only on the corresponding integral Weyl group and its action on the parameters of the Verma modules by giving a combinatorial description of the projective objects. As a n application we derive the Kazhdan-Lusztig conjecture for non–integral blocks from the integral case in finite and affine situations.

Character, Multiplicity, and Decomposition Problems in the Representation Theory of Complex Lie Algebras - Fiebig, Peter (Lie Groups, Differential Equations and Geometry. Advances and Surveys., 2017)

Kazhdan-Lusztig combinatorics via sheaves on Bruhat graphs - Fiebig, Peter (Contemporary Mathematics, 2007)
We study the projective objects in an exact category naturally associated to a Coxeter system. We discuss an analog of the Kazhdan-Lusztig conjecture and show how it follows from a "genericity" conjecture and how the latter follows from a "Hard Lefschetz" conjecture.

Lefschetz operators, Hodge-Riemann forms, and representations - Fiebig, Peter (International Mathematics Research Notices, 2020)
For a field of characteristic ̸= 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 weight lattice graded semisimple representation of the corresponding Lie alge- bra, 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 ̸= 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.

Localization of IC-complexes on Kashiwara's flag scheme and representations of Kac–Moody algebras - Carnovale, Giovanna; Esposito, Francesco; Fiebig, Peter (Indagationes Mathematicae, 2021)
We study equivariant localization of intersection cohomology complexes on Schubert varieties in Kashiwara's flag manifold. Using moment graph techniques we establish a link to the representation theory of Kac–Moody algebras and give a new proof of the Kazhdan–Lusztig conjecture for blocks containing an antidominant element.
CRIS-Team Scopus Importer:2021-08-06
Keywords: Intersection cohomology; Kac–Moody Lie algebra; Kashiwara thick flag variety; Moment graph

Lusztig's conjecture as a moment graph problem - Fiebig, Peter (Bulletin of the London Mathematical Society, 2010)
We survey a new approach towards Lusztig's conjecture on the irreducible characters of a reductive algebraic group over a field of positive characteristic. The main result that we review is that Lusztig's conjecture is implied by a multiplicity conjecture on the stalks of certain sheaves on a moment graph. This latter conjecture is known to hold if the underlying field is of characteristic 0. From this one can almost directly deduce the conjecture for fields of large enough characteristics; but using a Lefschetz-type theory on the moment graph, we can give an upper bound on the exceptions. Moreover, one can prove the multiplicity 1 case of the conjecture in full generality. In addition to a survey of the above results, we prove the equivalence between the original conjecture of Lusztig and its generic version, that is, the multiplicity conjecture for baby Verma modules for the corresponding Lie algebra. © 2010 London Mathematical Society.

Moment graphs in representation theory and geometry - Fiebig, Peter (Schubert Calculus - Osaka 2012, 2016)

On the restricted projective objects in the affine category O at the critical level - Fiebig, Peter (Contemporary Mathematics, 2012)
This article gives both an overview and supplements the articles [AF08] and [AF09] on the critical level category O over an affine Kac-Moody algebra. In particular, we study the restricted projective objects and review the restricted reciprocity and linkage principles.

On the restricted Verma modules at the critical level - Fiebig, Peter; Arakawa, Tomoyuki (Transactions of the American Mathematical Society, 2012)
We study the restricted Verma modules of an affine Kac-Moody algebra at the critical level with a special emphasis on their Jordan-Hölder multiplicities. Feigin and Frenkel conjectured 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. © 2012 American Mathematical Society.

On the subgeneric restricted blocks of affine category O at the critical level - Fiebig, Peter (Symmetries, Integrable Systems and Representations, 2013)
We determine the endomorphism algebra of a projective generator in a subgeneric restricted block of the critical level category O over an affine KacMoody algebra.© Springer-Verlag London 2013.

Parity sheaves, moment graphs and the p-smooth locus of Schubert varieties - Fiebig, Peter; Williamson, Geordie (Annales de l'Institut Fourier, 2014)
We show that the Braden-MacPherson algorithm computes the stalks of parity sheaves. 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 a (Kac-Moody) Schubert variety coincides with the rationally smooth locus, if the underlying Bruhat graph satisfies a GKM-condition.
Keywords: Constructible sheaves; Equivariant cohomology; Modular representation theory; Moment graphs; p-smooth locus; Schubert varieties; Tilting modules

PERIODICITY FOR SUBQUOTIENTS OF THE MODULAR CATEGORY 𝒪 - Fiebig, Peter (Transformation Groups, 2022)
In this paper we study the category O over the hyperalgebra of a reductive algebraic group in positive characteristics. For any locally closed subset K of weights, we define a subquotient O ([K]) of O . It has the property that its simple objects are parametrized by elements in K. We then show that O ([K]) is equivalent to O ([K +pl gamma]) for any dominant weight gamma if l > 0 is an integer such that K boolean AND (K - p(l)eta) = null for all weights eta > 0. Hence it is enough to understand the subquotients inside the dominant (or the antidominant) chamber.
CRIS-Team WoS Importer:2022-09-16

Quantum tilting modules over local rings - Fiebig, Peter (Journal of the London Mathematical Society-Second Series, 2023)
We show that tilting modules for quantum groups over local Noetherian domains of quantum characteristic 0 exist and the indecomposable tilting modules are parametrized by their highest weight. For this, we introduce a model category (Formula presented.) associated with a Noetherian (Formula presented.) -domain (Formula presented.) and a root system (Formula presented.). We show that if (Formula presented.) is of quantum characteristic (Formula presented.), the model category contains all (Formula presented.) -modules that admit a Weyl filtration. If (Formula presented.) is in addition local, we study torsion phenomena in the model category. This leads to a construction of torsion free, or “maximal” objects in (Formula presented.). We show that these correspond to tilting modules for the quantum group associated with (Formula presented.) and (Formula presented.).
CRIS-Team Scopus Importer:2023-01-27

Sheaves on affine Schubert varieties, modular representations, and Lusztig's conjecture - Fiebig, Peter (Journal of the American Mathematical Society, 2011)
We relate a certain category of sheaves of k-vector spaces on a complex affine Schubert variety to modules over the k-Lie algebra (for ch k>0) or to modules over the small quantum group (for ch k=0) associated to the Langlands dual root datum. As an application we give a new proof of Lusztig's conjecture on quantum characters and on modular characters for almost all characteristics. Moreover, we relate the geometric and representation theoretic sides to sheaves on the underlying moment graph, which allows us to extend the known instances of Lusztig's modular conjecture in two directions: We give an upper bound on the exceptional characteristics and verify its multiplicity one case for all relevant primes.

Sheaves on moment graphs and a localization of Verma flags - Fiebig, Peter (Advances in Mathematics, 2008)
To any moment graph <a class="mathText" href="http://www.sciencedirect.com/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870807002447&_mathId=si1.gif&_user=111111111&_pii=S0001870807002447&_rdoc=1&_issn=00018708&md5=c405cb7ae58379e166d6602e10e1ba65" title="View the MathML source">G we assign a subcategory <a class="mathText" href="http://www.sciencedirect.com/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870807002447&_mathId=si2.gif&_user=111111111&_pii=S0001870807002447&_rdoc=1&_issn=00018708&md5=7cbe07ff4aba593ff026ca71d34390e7" title="View the MathML source">V of the category of sheaves on <a class="mathText" href="http://www.sciencedirect.com/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870807002447&_mathId=si3.gif&_user=111111111&_pii=S0001870807002447&_rdoc=1&_issn=00018708&md5=6d70464e12a9a8898170f553ee034e92" title="View the MathML source">G together with an exact structure. We show that in the case that the graph is associated to a non-critical block of the equivariant category <a class="mathText" href="http://www.sciencedirect.com/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870807002447&_mathId=si4.gif&_user=111111111&_pii=S0001870807002447&_rdoc=1&_issn=00018708&md5=15ffa80d07ef7f08b36198bc2ac1d026" title="View the MathML source">O over a symmetrizable Kac–Moody algebra, <a class="mathText" href="http://www.sciencedirect.com/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870807002447&_mathId=si5.gif&_user=111111111&_pii=S0001870807002447&_rdoc=1&_issn=00018708&md5=fa5196e4cec25b6edfe88c0eb4429a0f" title="View the MathML source">V is equivalent (as an exact category) to the subcategory of modules that admit a Verma flag. The projective modules correspond under this equivalence to the intersection cohomology sheaves on the graph, and hence, by a theorem of Braden and MacPherson, to the equivariant intersection cohomologies of Schubert varieties associated to Kac–Moody groups.
Keywords: Sheaves on moment graphs; Koszul duality; Modules with Verma flag

Sheaves on the alcoves and modular representations I - Fiebig, Peter; Lanini, Martina (Transformation Groups, 2020)
We consider the set of affine alcoves associated with a root system R as a topological space and define a certain category S of sheaves of Zk-modules on this space. Here Zk is the structure algebra of the root system over a field k. To any wall reflection s we associate a wall crossing functor on S. In the companion article [FL] we prove that S encodes the simple rational characters of the connected, semisimple and simply connected algebraic group with root system R over k, in the case that k is algebraically closed with characteristic above the Coxeter number.

Sheaves on the alcoves and modular representations II - Fiebig, Peter; Lanini, M. (Transformation Groups, 2020)
We relate the category of sheaves on alcoves that was constructed in [FL1] to the representation theory of reductive algebraic groups. In particular, we show that its indecomposable projective objects encode the irreducible rational characters of a connected, semisimple and simply connected reductive algebraic group for characteristics above the Coxeter number.
CRIS-Team WoS Importer:2020-05-15

Subquotient categories of the affine category O at the critical level - Fiebig, Peter (Representation Theory - current Trends and Perspectives, 2017)

The combinatorial category of Andersen, Jantzen and Soergel and filtered moment graph sheaves - Fiebig, Peter; Lanini, Martina (Abhandlungen Aus Dem Mathematischen Seminar Der Universitat Hamburg, 2016)

The Combinatorics of Category O over symmetrizable Kac-Moody Algebras - Fiebig, Peter (Transformation Groups, 2006)
We show that the structure of a block outside the critical hyperplanes of category O over a symmetrizable Kac-Moody algebra depends only on the corresponding integral Weyl group and its action on the parameters of the Verma modules. This is done by giving a combinatorial description of the projective objects in the block. As an application, we derive the Kazhdan-Lusztig conjecture for nonintegral blocks from the integral case for finite or affine Weyl groups. We also prove the uniqueness of Verma embeddings outside the critical hyperplanes.

The combinatorics of Coxeter categories - Fiebig, Peter (Transactions of the American Mathematical Society, 2008)
We present an alternative construction of Soergel's category of bimodules associated to a reflection faithful representation of a Coxeter system. We show that its objects can be viewed as sheaves on the associated moment graph. We introduce an exact structure and show that Soergel's ``special'' bimodules are the projective objects. Then we construct the indecomposable projectives by both a global and a local method, discuss a version of the Kazhdan-Lusztig conjecture and prove it for universal Coxeter systems.

The linkage principle for restricted critical level representations of affine KacMoody algebras - Fiebig, Peter; Arakawa, Tomoyuki (Compositio Mathematica, 2012)
We study the restricted category ?' for an affine KacMoody 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 Bernstein"Gelfand" Gelfand-reciprocity principle and we determine the block decomposition of the restricted category ?'. For the proofs, we need a deformed version of the classical structures, so we mostly work in a relative setting. © 2012 Foundation Compositio Mathematica.
Keywords: linkage principle

The multiplicity one case of Lusztig's conjecture - Fiebig, Peter (Duke Mathematical Journal, 2010)
We prove the multiplicity one case of Lusztig's conjecture on the irreducible characters of reductive algebraic groups for all fields with characteristic above the Coxeter number. © 2010.

Zentren deformierter Darstellungskategorien und Verschiebungsfunktoren - Fiebig, Peter (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2001)
Wir definieren eine deformierte Version der klassischen Kategorie O über einer symmetrisierbaren Kac-Moody Algebra, untersuchen deren projektive Objekte und beweisen eine Version der klassischen BGG-Reziprozität. Lokalisierungstechniken erlauben die Berechnung der Zentren aller Blöcke außerhalb der kritischen Hyperebene. Darüber hinaus betrachten wir eine deformierte Version der Verschiebungsfunktoren in beide Richtungen und beweisen deren Adjungiertheit.