# Research Area A2

# Representations and symmetries in algebra and topology

## Research Area Leaders: Stefan Schwede, Catharina Stroppel

## PIs: Wolfgang Lück, Peter Scholze, Stefan Schwede, Catharina Stroppel

## Contributions by Pieter Belmans, Benjamin Böhme, Tobias Dyckerhoff (University of Hamburg), Thorsten Heidersdorf, Gustavo Jasso, Moritz Rahn, Jan Schröer, Geordie Williamson (University of Sydney)

## Topics and goals

This research area focuses on problems where modern topological tools are brought to bear on algebraic and representation-theoretic problems, and where refined algebraic structures and additional symmetries facilitate computations in topology. Cohomological invariants and categorification techniques are essential and connecting concepts. Hecke algebras and group representations appear in different disguises and are subject of our study through algebraic geometric, combinatorial and topological methods.

## State of the art, our expertise

**Equivariant homotopy theory and K-theory**. Schwede introduced a framework for global equivariant homotopy theory, i.e., spaces with simultaneous and compatible actions of all compact Lie groups, up to deformations that preserve all symmetries. He discovered the phenomenon called ultra-commutativity, a term used to describe a highly structured kind of equivariant multiplications, well beyond the classical notion of E1-structures, that includes transfer maps, norm maps, and power operations. This has opened the door to a rigorous study of global stable homotopy types, and global structures can now be exploited for new computations. A recent example where the global formalism uncovered a hitherto invisible universal description is Schwede’s calculation of equivariant stable homotopy groups of symmetric products [Sch17].

Chern characters for discrete proper action have been constructed by Lück. Already here a global point of view plays a role. Foundations of genuine equivariant homotopy theory for proper action of Lie groups were recently established by Lück, Schwede, and others. Groundbreaking work of Scholze and Nikolaus at the interface of equivariant stable homotopy theory and algebraic K-theory introduced a new formalism for cyclotomic spectra; this has led to a simplified and highly conceptual approach to topological cyclic homology.

**Canonical bases and Hecke algebras**. Dyckerhoff–Kapranov showed that Waldhausen’s S_-construction, originally introduced as a counterpart of Quillen’s Q_-construction for defining the higher algebraic K-groups, arises naturally in representation theory in the context of Hall algebras and Hecke algebras. He also found explicit formulas for A1-homotopy invariants of topological Fukaya categories of surfaces [Dyc17, DK], which opens the door to their combinatorial descriptions.

A Lie-theoretic viewpoint on certain Fukaya categories was given by Stroppel via an explicit description of categories of perverse sheaves [ES16], allowing geometric realizations of very different classical representation-theoretic categories [ES]. The existence of canonical bases in Hecke algebras is crucial here, as it is in Williamson’s groundbreaking work [EW14], which introduced a version of Hodge theory in the context of Soergel bimodules and yielded a proof ofthe famous Kazhdan–Lusztig conjectures for any Coxeter group. An algebraic proof of the even harder Jantzen conjectures seems to be in reach now thanks to Williamson’s local hard Lefschetz theorem and local Hodge–Riemann bilinear relations [Wil16]. For the study of semi-canonical bases, Schröer et al. [GLS17] developed a general framework which covers all symmetrizable, non-symmetric Kac–Moody algebras and gives a precise connection to cluster algebra combinatorics [GLS13].

**Representation theory of p-adic groups**. The representation theory of p-adic groups as GLn(Qp) in arbitrary characteristics is a challenging problem with connections to the representation theory of finite groups and groups of Lie type. It also plays a central role in the Langlands program and thus connects to RA A1. In particular, the local Langlands conjectures predict a classification of irreducible representations in terms of L-parameters. It was established for GLn by Harris– Taylor and Henniart in 2002 with a simplified proof by Scholze in [Sch13], but remains largely open for general reductive groups. On the other hand, there are several foundational questions purely on the representation-theoretic side, such as Bernstein’s conjecture that all supercuspidal representations are compactly induced, or questions about the fine structure of the blocks that lead to the study of Schur algebras. A new construction of these algebras in terms of geometric representation theory was recently found by Miemietz–Stroppel. If also the coefficients are p-adic, a theory of traces is missing. One open problem is whether one can use a modified cyclotomic trace to detect elements in algebraic K-theory of the Hecke algebra of a totally disconnected group, generalizing the approach for discrete groups of [LRRV17].

## Research program

**Equivariant homotopy theory and K-theory**. With the foundations for global structures and ultracommutative multiplications in place, the time is ripe to apply the formalism, for example to questions in homotopy theory and K-theory. The study of ultra-commutative ring spectra has just begun; some exciting and challenging problems are to compare global topological K-theory to the ultra-commutative localization along the Bott class of the tautological U(1)-representation, or to develop a global obstruction theory for ultra-commutative ring spectra. For the latter, global versions of basic algebraic concepts, such as étale extensions and André-Quillen homology, need to be developed from scratch in the context of global power functors. A more speculative long-term project is to investigate the universal properties of equivariant bordism from the global perspective, and to understand its connection to equivariant formal groups. Prominent isomorphism conjectures due to Farrell-Jones and Baum-Connes relate certain K- and L-groups to equivariant homology of proper G-spaces; we plan to extend the connection to new classes of groups such as totally disconnected groups or Kac-Moody groups. The focus and long-term goal of Lück and Schwede is the development of a genuine equivariant homotopy theory for actions of these groups; the applications to the Farell-Jones conjecture are part of RA A3. The computation of the equivariant homology groups for proper G-spaces should now take advantage of the sophisticated machinery. The global point of view hopefully allows us to implement additional structures such as transfers, products, and norm maps. Chern characters for totally disconnected groups shall be established and exploited for computations. The global algebraic K-theory spectrum of a commutative ring is a compact and rigid way of packaging the information that is contained in the representation K-theory of the ring, such as Swan groups of the group ring. Global algebraic K-theory supports an ultra-commutative multiplication, and our expectation is that it relates to one side of the isomorphism conjectures for infinite discrete groups.

**Canonical bases and Hecke algebras**. Determining the characters of indecomposable tilting modules for reductive groups is one of the most fundamental open problems in modular representation theory. A solution for GLn would answer the question of the dimensions of the simple modules for the symmetric group in characteristic p. It is related to - but almost certainly harder than - the determination of the simple characters. A long-term goal of Williamson and Stroppel is to develop theoretical as well as algorithmic tools to attack this problem. There are many ideas involved, ranging from geometric representation theory and the representation theory of monoidal categories to combinatorial aspects. The easier, but still poorly understood representation theory of algebraic supergroups over the complex numbers will serve as a test case. A nontrivial generalization of Lusztig's canonical basis theory is necessary, in particular in light of Williamson's disproof of the famous Lusztig conjectures. By bringing together higher-categorical and simplicial methods with recent developments in representation theory of finite-dimensional algebras, Dyckerhoff and Jasso seek to develop higher-dimensional analogues of Waldhausen's S_-construction and their higher-categorical and combinatorial properties. In analogy to the original construction, which is closely related to the representation theory of linearly oriented quivers of type A, its higher-dimensional analogues are related to the representation theory of certain quivers with relations which have been independently investigated in the representation theory of finite-dimensional algebras in the context of Iyama's higher Auslander-Reiten theory. Auslander-Reiten theory is also a crucial ingredient in Schröer's long-term goal of developing from scratch a new treatment of the representation theory of wild quivers. This will provide a universal approach to wild finite-dimensional algebras that contain all module categories of all finite-dimensional algebras via suitable embedding functors. Continuing the collaboration with Geiss and Leclerc [GLS17], Schröer will refine the description of the semicanonical bases in terms of cluster algebra structures related to Kac-Moody algebras. On the other hand, topological Fukaya categories, introduced by Dyckerhoff, promise to give interesting applications to cluster categories of surfaces and their categorifications. Dyckerhoff, Schröer, and Stroppel will work on different aspects connecting Fukaya categories and cluster algebras with topological invariants of manifolds and representation theory via categorification techniques. In IRU D2, similar problems in categorification arise and both areas benefit from the interaction.

**Representation theory of p-adic groups**. The analogue of the Farrell-Jones conjecture for the Hecke algebra H(G) of a reductive p-adic group G in terms of algebraic K-theory is a connecting topic with RA A3. A question in Lück's research agenda is whether the projective class group of the Hecke algebra H(G) can be reconstructed from that of its compact open subgroups. This is particularly important for the understanding of smooth representations since they often admit finite projective resolutions when considered as modules over the Hecke algebra, and thus define elements in the projective class group. We expect a deeper insight into the Bernstein conjecture via the envisioned proof of the Farrell-Jones conjecture for the algebraic K-theory of the Hecke algebra.

Using generalizations of Khovanov-Lauda-Rouquier algebras, Stroppel pursues a description of the fine structure of blocks for GLn(Qp) over fields of characteristics coprime to p including homological properties, natural gradings and character formulas. Scholze wants to construct the L-parameters associated to an irreducible representation of G(Qp) for a general reductive group G, using spaces of mixed-characteristic shtukas from RA A1 generalizing Lubin-Tate spaces and Rapoport-Zink spaces building on [SW13]. Complementary to RA A1, Scholze's focus here is on the representation theory, including the detailed study of cuspidal L-packets and the Jacquet- Langlands transfer. Recently, Fargues conjectured a geometrization of the local Langlands correspondence which assembles this information into a geometric Langlands-type conjecture on the space of G-bundles on the Fargues-Fontaine curve. This makes it possible to import techniques from geometric representation theory. Even for GLn new consequences from this picture are expected, and the Harris-Viehmann conjecture and the Kottwitz conjecture on the cohomology of Rapoport-Zink space are within reach. With p-adic coefficients, Scholze wants to investigate whether topological cyclic homology and the cyclotomic trace provide a good theory of traces.

## Summary

This research area combines our expertise in algebraic topology, representation theory and arithmetic geometry. The ambitious, yet realistic, aim is to make substantial progress on several prominent conjectures in these fields, and investigate deeper and yet unexplored connections. Representations of different kinds of groups, such as Lie groups, algebraic groups, and finite groups of Lie type, are studied from complementary perspectives. The unifying theme is to uncover and exploit the hidden geometry behind previously isolated phenomena.

## Bibliography

[DK] T. Dyckerhoff and M. Kapranov. Triangulated surfaces in triangulated categories.* J. Eur. Math. Soc. (JEMS). to appear*.

[Dyc17] T. Dyckerhoff. A1-homotopy invariants of topological Fukaya categories of surfaces. *Compos. Math., 153(8):1673–1705, 2017*.

[ES] M. Ehrig and C. Stroppel. Nazarov–Wenzl algebras, coideal subalgebras and categorified skew Howe duality.* Adv. Math. to appear*.

[ES16] M. Ehrig and C. Stroppel. Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians. Selecta Math. *(N.S.), 22(3):1455–1536, 2016*.

[EW14] B. Elias and G. Williamson. The Hodge theory of Soergel bimodules. *Ann. of Math. (2), 180(3):1089–1136, 2014*.

[GLS13] C. Geiß, B. Leclerc, and J. Schröer. Cluster structures on quantum coordinate rings. *Selecta Math. (N.S.), 19(2):337–397, 2013*.

[GLS17] C. Geiss, B. Leclerc, and J. Schröer. Quivers with relations for symmetrizable Cartan matrices I: Foundations.* Invent. Math., 209(1):61–158, 2017*.

[LRRV17] W. Lück, H. Reich, J. Rognes, and M. Varisco. Algebraic K-theory of group rings and the cyclotomic trace map. *Adv. Math., 304:930–1020, 2017*.

[Sch13] P. Scholze. The local Langlands correspondence for GLn over p-adic fields.* Invent. Math., 192(3):663–715, 2013*.

[Sch17] S. Schwede. Equivariant properties of symmetric products.* J. Amer. Math. Soc., 30(3):673– 711, 2017*.

[SW13] P. Scholze and J. Weinstein. Moduli of p-divisible groups.* Camb. J. Math., 1(2):145–237, 2013*.

[Wil16] G. Williamson. Local Hodge theory of Soergel bimodules. *Acta Math., 217(2):341–404, 2016*.