Peter Fiebig

An upper bound on the exceptional characteristics for Lusztig's character formula

Abstract We develop a Lefschetz theory in a combinatorial category associated to a root system and derive an upper bound on the exceptional characteristics for Lusztigs formula for the simple rational characters of a reductive algebraic group. Our bound is huge compared to the Coxeter number.

Sheaves on affine Schubert varieties, modular representations, and Lusztig's conjecture

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 Lusztigs 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 Lusztigs modular conjecture in two directions: We give an upper bound on the exceptional characteristics and verify its multiplicity one case for all relevant primes.

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 Bernstein–Gelfand–Gelfand-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.

The multiplicity one case of Lusztig's conjecture

We prove the multiplicity one case of Lusztigs conjecture on the irreducible characters of reductive algebraic groups for all fields with characteristic above the Coxeter number.

On the Subgeneric Restricted Blocks of Affine Category \(\mathcal{O}\) at the Critical Level

We determine the endomorphism algebra of a projective generator in a subgeneric restricted block of the critical level category \(\mathcal{O}\) over an affine Kac–Moody algebra.

Character, Multiplicity, and Decomposition Problems in the Representation Theory of Complex Lie Algebras

This essay is meant as an introduction to a very successful approach towards understanding the structure of certain highest weight categories appearing in algebraic Lie theory. For simplicity, the focus lies on the case of the category \(\mathcal{O}\) of representations of a simple complex Lie algebra. We show how the approach yields a proof of the classical Kazhdan–Lusztig conjectures that avoids the theory of D-modules on flag varieties.