“Hendoff” for inaugural SMRI Exec Director Anthony Henderson: 19–22 April 2022

Abstract: A metric on a category assigns lengths to morphisms, with the triangle inequality holding. This notion goes back to a 1974 article by Lawvere. We’ll begin with a quick review of some basic constructions, like forming the Cauchy completion of a category with respect to a metric.

And then will begin a string of surprising new results. It turns out that, in a triangulated category with a metric, there is a reasonable notion of Fourier series, and an approximable triangulated category can be thought of as a category where many objects are the limits of their Fourier expansions. And then come two types of theorems: (1) theorems providing examples, meaning showing that some category you might naturally want to look at is approximable, and (2) general structure theorems about approximable triangulated categories.

And what makes it all interesting is (3) applications. These turn out to include the proof of a conjecture by Bondal and Van den Bergh, a major generalization of a theorem of Rouquier’s, a short, sweet proof of Serre’s GAGA theorem and a proof of a conjecture by Antieau, Gepner and Heller.

Abstract: Admissible representations of real reductive groups are a key player in the world of unitary representation theory. The characters of irreducible admissible representations were described by Lustig—Vogan in the 80’s in terms of a geometrically-defined module over the associated Hecke algebra. In this talk, I’ll describe a categorification of a block of the LV module using Soergel bimodules.

Abstract: It is well known that the geometry of Bruhat-Tits buildings encodes many interesting algebraic properties of the corresponding groups. Retractions play a key role in this. We introduce chimney retractions for arbitrary affine buildings and give a recursive description of the shadows which they induce. We then explain how these shadows can be used to study subvarieties of and orbits in (partial) affine flag varieties. This is joint work with Elizabeth Milićević, Yusra Naqvi and Petra Schwer.

Abstract: We investigate the 𝑖-quantum groups 𝑈^𝐽(n) and 𝑈^𝑖(n) and their associated q-Schur algebras 𝑺^𝐽(n,r) and 𝑺^𝑖(n,r) of types B and C, respectively. This includes short (element) multiplication formulas, long (element) multiplication formulas, and triangular relations in 𝑺^𝐽(n,r) and 𝑺^𝑖(n,r). We will also give new realisations of Beilinson–Lusztig–MacPherson type for both 𝑈^𝐽(n) and 𝑈^𝑖(n) and discuss their Lusztig forms. This allows us to link representations of 𝑈^𝐽(n) and 𝑈^𝑖(n) with those of finite orthogonal and symplectic groups.

This is joint work with Yadi Wu.

Abstract: I will explain how mixed Hodge modules can be utilized to understand representation theory of real groups. In particular, we obtain a refinement of the Lusztig-Vogan polynomials in this setting. Adams, van Leeuwen, Trapa, and Vogan (ALTV) have given an algorithm to determine the unitary dual of a real reductive group. As a corollary of our results we obtain a proof of a key result of (ALTV) on signature polynomials. This is joint work with Dougal Davis.

Abstract: Recent results in algebraic geometry, as well as in the theory of tensor categories, motivate studying the process of taking the inverse limit of an affine group scheme (over a field of positive characteristic) along the Frobenius homomorphism. This is the ‘perfection’ of the group scheme.
I will focus mainly on the perfection of reductive groups. In particular, I will discuss their classification in combinatorial terms, the relation with topological localisation of classifying spaces and with generic cohomology.
This is joint work with Geordie Williamson.

Abstract: Character variety of surface groups plays a central role in diverse areas of mathematics such as Geometric Langlands program and non-abelian Hodge theory. Determining cohomology of the character variety has been a subject active research for decades. In this talk, I will report an on-going project to count points on character varieties over finite fields. The main goal is to generalise the work of Hausel—Letellier-Villegas from type A to arbitrary type. A key role is played by representation theory of finite reductive groups (Deligne—Lusztig theory, Lusztig’s Jordan decomposition, etc.)

Abstract: The Exotic Springer correspondence as defined by Kato provides a bijection between $Sp$-orbits on the Hilbert nullcone of a certain $Sp_{2n}$-module and irreducible representations of the Weyl group of type C, $W(C_n)$. This particular nullcone is known as the Exotic nilpotent cone. With V. Nandakumar and D. Rosso, we demonstrated an explicit bijection between irreducible components of the fibres of a resolution of the exotic nilpotent cone and standard Young bitableaux, which label the irreducible representations of $W(C_n)$ . In this talk, we will detail some results that follow from this bijection and report on ongoing work with D. Rosso in extending these results to partial flag varieties in type C.

Abstract: I will talk about the folding process in categorification, which allows one to extend results from simply-laced Lie algebras to general
symmetric type, or from Hecke algebras with equal parameter to Hecke algebras with unequal parameters. Particular attention will be given to the categorification of the cluster structure on modules over KLR algebras, where we can prove that every cluster monomial lies in the dual (p-)canonical basis.

Abstract: Let $G$ be a connected reductive group over an algebraically closed field, and let $C$ be a nilpotent orbit for $G$. If $L$ is an irreducible $G$-equivariant vector bundle on $C$, then by work of Deligne, Bezrukavnikov, Arinkin, one can define a “coherent intersection cohomology complex” $IC(C,L)$. These objects play an important role in various results related to the local geometric Langlands program.
When $G$ has positive characteristic, instead of an irreducible bundle $L$, one might consider a tilting bundle $T$ on $C$. I will explain a new construction that associates to the pair $(C,T)$ a complex of coherent sheaves $S(C,T)$ with remarkable Ext-vanishing properties. This construction leads to a proof of a conjecture of Humphreys on (relative) support varieties for tilting modules. This work is joint with W. Hardesty (and also partly with S. Riche).

Abstract: We discuss character sheaves in the setting of graded Lie algebras, concentrating on the construction of cuspidal character sheaves. Irreducible representations of Hecke algebras of complex reflection groups at roots of unity enter the description of character sheaves. We will explain the connection between our work and the recent work of Lusztig and Yun, where irreducible representations of trigonometric double affine Hecke algebras appear in the picture. This is based on joint work with Kari Vilonen and partly with Misha Grinberg.

Abstract: The 2-Calabi-Yau category of a quiver is a basic object of interest in study of braid groups and their higher representation theory. In this talk I’ll describe how to extract some geometry from this object, by endowing the set of spherical objects of the category with a topology. Conjectures (and, less prominently, theorems) will appear.

Abstract: In this talk, I will introduce a finite dimensional algebra, called noncrossing algebra, associated with the noncrossing partition lattice of a finite Coxeter group. This new algebra has some similarities with the Orlik-Somolon algebra, but involves new combinatorics in its construction. I will show how this algebra can be used to compute the cohomology of the Milnor fibre of the corresponding Coxeter arrangement. This is joint work with Gus Lehrer.

Abstract: The Baker-Campbell Hausdorff Theorem gives a formula for log(e^xe^y) where x and y don’t commute. The Kashiwara-Vergne theorem is a refinement of this fact, and has wide implications in harmonic analysis and Lie theory: in particular, it implies the multiplicativity of the Duflo isomorphism. There are two different topological approaches to the Kashiwara-Vergne theorem and its cousins: one via four-dimensional knot theory (joint works with Dror Bar-Natan, Iva Halacheva, Tamara Hogan, Marcy Robertson and Nancy Scherich), and one via homotopy curves on surfaces due to Alekseev, Kawazumi, Kuno and Naef.
This talk is an overview of these approaches, their implications, and open questions. Thank you, Anthony, for encouraging me to come to Sydney, and for your support and mentorship over the years.