# Representation theory seminar 2023 Semester 2

**Organisers**: Dougal Davis, Kari Vilonen, Ting Xue

**Place and time: **Peter Hall 162, Thursdays 3:15 – 5:00pm

Please contact one of the organisers to be added to the mailing list.

October 5: **Dougal Davis** (University of Melbourne) Mixed Hodge modules and real groups I

The aim of this lecture series is to explain recent progress of myself and Kari Vilonen on the deep problem of determining the unitary representations of a real reductive Lie group. Our main theorem is that unitary representations are completely governed by Hodge theory on the flag variety (originally conjectured by Schmid and Vilonen over 10 years ago). In these lectures, which are supposed to be accessible to students, I will explain our results (and a bit about the proofs), as well as some of the necessary background on real groups, Hodge theory and Beilinson-Bernstein localisation.

Reference: D. Davis and K. Vilonen, Unitary representations of real groups and localization theory for Hodge modules.

In the first lecture, I will give an introduction to the unitary representation theory of real groups, focusing on the example of . I will try to give some idea of the shape of what is known in general, and how our work on Hodge theory enters this picture.

October 12: **Dougal Davis** (University of Melbourne) Mixed Hodge modules and real groups II

In this second lecture, which is a geometric interlude, I will give an introduction to the general theory of mixed Hodge modules. I will recall some basic results in classical Hodge theory, explain how Hodge modules are supposed to generalise this, and touch on some of the most important technical aspects for our story.

October 19: **Dougal Davis** (University of Melbourne) Mixed Hodge modules and real groups III

The aim of this third lecture is to state the main results from our recent paper. The starting point is the classical Beilinson-Bernstein localisation theorem, which relates Lie algebra representations to D-modules on the flag variety. I will briefly recall how this goes and explain how the statements can be refined from D-modules to mixed Hodge modules. I will also explain the full conjecture of Schmid and Vilonen (which is still a conjecture) and the closely related Hodge-theoretic criterion for unitarity (which we have proved).

October 26: **Dougal Davis** (University of Melbourne) Mixed Hodge modules and real groups IV

In this fourth and final lecture, I will explain some of the ingredients in the proofs of our main theorems.

### Past Seminars

September 14: **Arun Ram** (University of Melbourne) Introduction to Hessenberg varieties and relations to polynomials from combinatorics

Learning seminar on Hessenberg varieties, talk 3

I will define Hessenberg varieties and give some examples. Then I will explain some of the “combinatorial” results (Shareshian-Wachs, Brosnan-Chow, Abe-Horiguschi-et al, Precup-Sommers) about the cohomology of Hessenberg varieties coming from b-submodules of the adjoint representation. I will endeavour to explain the structural/conceptual relationship between this “non-interesting” setting and the “interesting” setting that Yau Wing has introduced us to. As Kari explained, the “interesting” setting is when the motive of the Hessenberg variety shows features that do not appear in the Springer resolution and the Grothendieck simultaneous resolution. For this talk I will move away from the Springer resolution and the Grothendieck simultaneous resolution but I will stick to the very fascinating and amazing “non-interesting” setting.

September 7:

Learning seminar on Hessenberg varieties, talk 2

**Yau Wing Li **(University of Melbourne) On the cohomology of Fano varieties and the Springer correspondence, continued

**Dougal Davis and Kari Vilonen** (University of Melbourne) Purity of equivalued affine Springer fibers

We will make some comments on the paper of the same title by Goresky, Kottwitz and MacPherson.

August 31: **Yau Wing Li** (University of Melbourne) On the cohomology of Fano varieties and the Springer correspondence

Learning seminar on Hessenberg varieties, talk 1

I will discuss the paper with the same title by Tsao-Hsien Chen, Kari Vilonen and Ting Xue. The main result is a computation of the cohomology of the Fano varieties of k-planes inside a smooth intersection of quadrics. The computation works by realising these as certain Hessenberg varieties and relating their cohomology to Springer theory for symmetric spaces by a Fourier transform.

Reference: T.-H. Chen, K. Vilonen, T. Xue, On the cohomology of Fano varieties and the Springer correspondence.

August 17: **Ryo Fujita **(Kyoto University) Isomorphisms among quantum Grothendieck rings and their cluster theoretical interpretation

Quantum Grothendieck ring in this talk is a one-parameter deformation of the Grothendieck ring of the monoidal category of finite-dimensional modules over the quantum loop algebras, endowed with the canonical basis consisting of simple (q,t)-characters. In the case of type ADE, thanks to Nakajima’s geometric theory of quiver varieties, these simple (q,t)-characters are known to compute the q-characters of simple modules (via the analog of Kazhdan-Lusztig algorithm) and enjoy some positivity properties. In this talk, we discuss a collection of isomorphisms between the quantum Grothendieck ring of type BCFG and that of “unfolded” type ADE, which respect the canonical bases. They are applied to verify the same positivity properties in type BCFG and the analog of Kazhdan-Lusztig conjecture for several new cases. We also discuss their cluster theoretical interpretation, which particularly yields non-trivial birational relations among the (q,t)-characters of different types. This is a joint work with David Hernandez, Se-jin Oh, and Hironori Oya.

July 20: **Shigenori Nakatsuka** (University of Alberta) Recent progress in the dualities of W-algebras

*Place and time*: Peter Hall 162, 3:15pm – 5:00pm

W-algebras provide a rich family of vertex algebras parametrized by simple Lie algebras and their nilpotent orbits. They are vertex algebraic analogue of associative algebras, called the finite W-algebras, interpolating the enveloping algebras and their centers. The principal W-algebras enjoy a nontrivial isomorphism called the Feigin-Frenkel duality, which can be seen as the upgrade of the isomorphism between the centers for the Langlands dual simple Lie algebras. In this talk, I will explain the recent progress on our understanding of the dualities of W-algebras motivated by physics and some consequences for their representation theory.

July 27: **Wille Liu **(Academia Sinica) Translation functors for trigonometric double affine Hecke algebras

*Place and time*: Peter Hall 162, 3:15pm – 5pm

Double affine Hecke algebras were introduced by Cherednik around 1995 as a tool to study the Macdonald polynomials. The trigonometric double affine Hecke algebras (TDAHA), degenerate version of the former, have also been found related to several other areas. In this talk, I will be focusing on specific aspects of the representation theory of the TDAHA.

Given a root system, the TDAHA depends on a family of complex parameters . Given two families of parameters and such that takes values in , there exists an equivalence of derived categories of the corresponding TDAHA: , called translation functor. After a brief introduction to the TDAHA, I will talk about a construction of translation functors.

August 3: **Drazen Adamovic **(University of Zagreb) On the semi-simplicity of the category KL for affine vertex algebras at collapsing levels

*Place and time*: Peter Hall 162, 3:15pm – 4:15pm

In this talk, we will report on recent results on the representation theory of the simple affine vertex algebra at collapsing levels. When is an admissible rational number and is a Lie algebra, then each –module in the category is completely reducible by a result of Arakawa. But it turns out that the Kazhdan-Lusztig category of –modules can be also semi-simple for non-admissible levels . We will present an approach which uses the representation theory of minimal affine -algebras and the quantum Hamiltonian reduction functor. We proved this in joint papers with V. Kac, P. Moseneder Frajria, P. Papi and O. Perse that is semi-simple when the simple affine -algebra is rational or when is collapsing level for . This result enables us to prove the complete reducibility of modules in for some non-admissible levels.

In the case when is a Lie superalgebra, the analysis of the category is more delicate than in the Lie algebra case, since in the super case can contain indecomposable modules. But even in this case we have a complete reducibility result for collapsing levels.

August 9: **Anne Dranowski** (USC) ***** Extra seminar: note unusual day (Wednesday)! *****

*Place and time*: Peter Hall 162, 3:15pm – 4:15pm

**gl(2) webs, foams and spectra**

Webs were introduced by Kuperberg to construct sl(3) link invariants and foams were introduced by Khovanov to construct sl(3) link homology. Blanchet’s oriented and {1,2}-labeled version of Khovanov’s foam category fixed the functoriality of Khovanov’s categorification over the integers. Via the extra info encoded in the labels it also offered a sl to gl upgrade. Relying on Blanchet’s foams and Howe duality, Lauda, Queffelec and Rose realized foams as representations of . In another direction, Lawson, Lipshitz and Sarkar constructed a stable homotopy refinement of Khovanov homology. Joint work in progress with Guo, Lauda and Manion pieces together these results to construct spectral bimodules for . In his talk we review what’s known, as well as what’s suggested, and how you might prove it.

August 10: **Cheng-Chiang Tsai** (Academia Sinica) Wave-front sets for p-adic groups and graded Springer theory

*Place and time*: Peter Hall 162, 3:15pm – 5pm

For an irreducible admissible representation of a p-adic reductive group there is the notion of its wave-front set, which is a set of nilpotent orbits that describes the asymptotic behavior of the character near the identity. By a theorem of Moeglin-Waldspurger, the set describes the least degenerate Whittaker models, which are double generalizations of local components of Fourier expansions for modular forms.

In this talk, we explain how a significant part of the determination of wave-front sets can be translated into Lie-theoretic questions regarding generalizations of Kostant sections, which can be related to graded Springer theory. This last viewpoint has led us to discover surprising behaviors of wave-front sets for p-adic groups that do not occur for real-groups, disproving some long-standing conjectures.

(We do not assume any familiarity with graded Springer theory, or even Springer theory.)