Seminar Darstellungstheorie
Das Seminar Darstellungstheorie findet normalerweise dienstags, 14.15 - 15.15 Uhr in G.13.18 statt.
Beteiligte Dozenten: Sascha Orlik, Kay Rülling, Tobias Schmidt, Britta Späth.
Zhixiang Wu: Bernstein--Zelevinsky duality for locally analytic principal series representations
Bernstein--Zelevinsky duality is classically a duality on the derived category of smooth representations of a p-adic Lie group. In this talk, we will consider the Bernstein--Zelevinsky duality for locally analytic representations of p-adic Lie groups, and compute explicitly the duality for principal series representations. This is joint work with Matthias Strauch.
Erik Barinaga: On Chow groups of Deligne--Lusztig varieties
Deligne--Lusztig varieties play a very important role in the representation theory of finite groups of Lie type, as covered in the landmark 1976 paper by Deligne and Lusztig. Nevertheless, these schemes remain mysterious in general. For instance, as of yet we have no real strategy to determine their l-adic cohomology groups in a coherent way, especially outside of the general linear case, or small rank examples. Likewise, not much is known about their Picard groups or more general (higher) Chow groups. In this talk, we will present some results on (rational, higher) Chow groups of Deligne--Lusztig varieties. We discuss a recursive formula describing the rational Picard group of a Deligne--Lusztig variety in terms of a smaller variety in the case of G=Gl_n. If time permits, we also discuss the integral Picard group of some fixed Weyl group elements. This is work in progress, and part of my PhD thesis.
James Taylor: Categories of Equivariant Vector Bundles on Drinfeld Symmetric Spaces
In this talk we will describe various categories of equivariant vector bundles and vector bundles with connection on Drinfeld symmetric spaces. These include the category of G^0-finite ("torsion") vector bundles with connection and the category of Lubin-Tate bundles introduced by Kohlhaase.
Tommy Lundemo: On the residue sequence in logarithmic THH
While algebraic K-theory, TC, and THH all enjoy a common localization property, a key tool to study the resulting cofiber sequences - dévissage - is only available for algebraic K-theory. There have been two major attempts to circumvent this: The first, due to Hesselholt--Madsen and Blumberg--Mandell, involves a model of THH and TC of Waldhausen categories that produces new localization sequences as an instance of Waldhausen's fibration theorem. The second, due to Rognes--Sagave--Schlichtkrull, bypasses dévissage entirely. Instead, they use Rognes' logarithmic THH to generalize a classical residue sequence involving logarithmic differential forms to a cofiber sequence of cyclotomic spectra.
I will report on work in progress, in which I will address the problem of reconciling the localization property of THH with that of its logarithmic counterpart. When combined with forthcoming work of Ramzi--Sosnilo--Winges, this becomes closely related to the problem of realizing logarithmic THH as the THH of a stable infinity-category, providing a candidate category of "logarithmic modules" in specific cases.
Ting-Han Huang: On the p-adic Gross–Zagier formula for triple product p-adic L-functions attached to finite slope families
We will briefly explain the work in the thesis (and a recent paper with A. Kazi) of the speaker, which revolves around the p-adic Gross–Zagier formula for (twisted) triple product p-adic L-functions attached to finite slope families. Such a formula relates the values outside the interpolation range to certain p-adic Abel–Jacobi map images, and serves as a crucial ingredient to study the Bloch–Kato conjecture or special case of BSD-conjecture. We will skip the geometric part of the story and focus instead on the computation of the polynomial q-expansions. In the end, we will demonstrate the relation between the special values of p-adic L-functions and the p-adic Abel–Jacobi images of diagonal cycles.
Arun Soor: Relation between D-cap modules and quasi-coherent sheaves on the de Rham space
In algebraic geometry, there is an equivalence between the category of D-modules (on a smooth scheme X) and the category of quasi-coherent sheaves on a certain stack X_dR associated to X.
I will explain an analogous relation in rigid-analytic geometry. Namely, I will explain how to produce a fully-faithful functor from the category of coadmissible D-cap modules of Ardakov-Wadsley (more generally, the C-complexes of Bode) to the category of quasi-coherent sheaves on the “analytic de Rham space”.
Annika Bartelt: Character degrees in principal blocks for distinct primes
A conjecture of Navarro, Rizo and Schaeffer Fry (2022) relates the absence of certain characters of a finite group to the existence of a certain subgroup. The characters in question should lie in both principal blocks for two distinct primes p and q and have degree coprime to p and q. The authors reduced their conjecture to a problem on almost simple groups and proved it in some cases. In this talk, we consider the open case of simple classical groups of Lie type (and non-defining distinct (odd) primes p and q), and examine certain unipotent characters. This work is part of the speaker's PhD thesis and based on arXiv:2506.06860.
Fernando Pena Vazquez: Hochschild (co)-homology of D-modules on rigid analytic spaces
The Hochschild homology and cohomology of an associative algebra A are fundamental invariants which store a substantial amount of information on the algebraic properties of A . For instance, Hochschild homology can be regarded as the derived trace space of A, and leads to more refined invariants such as cyclic homology and periodic cyclic homology. Due to the work of A. Connes, we know that these invariants can be used to obtain a generalization of the de Rham cohomology for non-commutative algebras. Similarly, the Hochschild cohomology of A can be regarded as its derived center, and it is closely related to the deformation theory of A. Let X be a smooth and separated rigid analytic space over a complete non-archimedean field of mixed characteristic. In this talk, we will introduce the formalism of Hochschild (co)-homology for D-modules on X, and use it to study the deformation theory of the algebra of completed p-adic differential operators on X. In order to have a good setting in which to perform homological algebra, we will work within the framework of sheaves of Ind-Banach D-modules and quasi-abelian categories. An upshot of this construction is that the corresponding (co)-homology spaces store the same information as their algebraic counterparts, but also remember the analytical structure of the algebra of completed p-adic differential operators on X.
