Representation Theory Days
(in honor of George Lusztig)

This is about joint work with Jeffrey Adams, Stephen Miller, and Marc van Leeuwen.

I'll formulate a conjecture (the "FPP conjecture") about the nature of the unitary dual for a general real reductive G. This conjecture originates in the work of Dan Barbasch in the late 1980s; a great deal of supporting evidence can be found in the work of Barbasch and his collaborators since that time.

The FPP conjecture defines for each real reductive G a finite collection of compact polyhedra FPP_j(G), each of dimension at most the semisimple rank of G. The entire unitary dual is (according to the conjecture) a countable union of pieces, each isomorphic to some FPP_k(L) for an appropriate Levi subgroup L of G.

For SL(2,R) there are exactly six of these polyhedra: five points (the first two discrete series, the two limits of discrete series, and the reducible unitary principal series) and one interval [0,1] (the spherical complementary series).

The atlas software is able in principle to calculate all the polyhedral FPP_j(G). For the split real form of E7, the number of polyhedra is 2025524; computing all of them, as Jeff Adams has done, requires a few months of CPU time, using a few hundred gigabytes of memory.

I hope to finish by explaining what all these gigabytes and CPUs have suggested to us about how to prove the FPP conjecture.

Abstract

Let $G$ be a finite reductive group over a field of characteristic $p$. In the talk I will survey recent results in the $\ell$-modular representation theory of $G$ for primes $\ell\ne p$ and will try to illustrate how perfectly, or even miraculously, Lusztig's theory of characteristic zero representations and Brauer's modular theory fit together. Mainly, this is joint work with Radha Kessar.

If, from the axioms defining tannakian categories, one omits the existence of fibers functors, one defines pretannakian categories. For a long time, the only examples were obtained by interpolation from categories of representations of algebraic groups or super groups. In recent years, wholly new very interesting examples were found (works of A. Snowden, N. Harman, S.Kriz, ...).

In a 1994 paper, Lusztig initiated the study of sheaves on the (enhanced) affine flag variety with a fixed monodromy along torus orbits. One feature of his setup is that he allows nontrivial monodromy along the Kac-Moody central torus, a precursor of the "metaplectic" or "quantum" geometric Langlands setting. In joint work with Gurbir Dhillon, Yau-Wing Li and Xinwen Zhu, we show that these monodromic affine Hecke categories are equivalent to the more familiar non-monodromic affine Hecke categories of smaller groups. The construction of these smaller groups is an affine analogue of the notion of endoscopic groups defined by Langlands. Our equivalence generalizes previous work of Lusztig and myself for finite Hecke categories, and has several applications to the metaplectic/quantum Langlands program.

Roughly speaking, an affine Deligne-Lusztig variety describes the intersection of a given Iwahori double coset with a Frobenius-twisted conjugacy class in the loop group; while an affine Lusztig variety describes the intersection of a given Iwahori double coset with an ordinary conjugacy class in the loop group. The affine Deligne-Lusztig varieties provide a group-theoretic model for the reduction of Shimura varieties and play an important role in the arithmetic geometry and Langlands program. The affine Lusztig varieties encode the information of the orbital integrals of Iwahori-Hecke functions and serve as building blocks for the (conjectural) theory of affine character sheaves. In this talk, I will explain a close relationship between affine Lusztig varieties and affine Deligne-Lusztig varieties, and consequently, provide an explicit nonemptiness pattern, dimension formula and enumeration of irreducible components for affine Lusztig varieties in many cases.

6 years ago Lusztig formulated a number of conjectures on the reduction mod $p$ of complex representations of finite reductive group. I will present some results (a joint work with R.Bezrukavnikov, C.Morton Ferguson, and M.Finkelberg) related to Lusztig's conjectures.

We show a number of examples where computer algebra methods and, especially, the CHEVIE system have been helpful in the theory of reductive algebraic groups. This concerns a whole range of topics, including conjugacy classes of Weyl groups, Kazhdan-Lusztig cells, unipotent classes of simple algebraic groups, the Green functions of Deligne-Lusztig and characters of finite groups of Lie type.

Lusztig introduced the notion of special nilpotent/unipotent orbits in a simple Lie algebra/group. To quote from one of his eight 1997 papers: "They play a key role in several problems in representation theory, such as the classification of irreducible complex representations of a reductive group over a finite field, and the classification of primitive ideals in the enveloping algebra of a semisimple Lie algebra." This talk will survey some results related to special nilpotent orbits and their associated special pieces. In particular, we will discuss the resolution of a conjecture of Lusztig posed in that 1997 paper: every special piece is the quotient of a smooth variety by a certain finite group (introduced in the paper). For classical groups, the result had already been established by Kraft and Procesi. With Juteau, Levy, and Yu, we establish the result for the exceptional groups and give a new proof for the classical groups.

I will explain how a number of quantum group constructions arise from categorical, homological and homotopical considerations. These come from corresponding structures on 2-representations of Kac-Moody algebras.

The flag variety of a reductive group is a projective algebraic variety with as many independent $P^1$-fibrations as the rank of its Neron-Severi group. I'll explain that the converse is true.

Let $G$ be either the symplectic group $Sp(V)$ or the special orthogonal group $SO(V)$ and the centralizer $G^{\iota}$ of a 1-parameter subgroup $\iota : \mathbb{C}^{*} \rightarrow G$. The group $G^{\iota}$ acts on the subspace $Lie_{2}(G)$ of the Lie algebra $Lie(G)$ on which $Ad(\iota(t))$ acts as $t^{2}$ times the identity for any $t \in \mathbb{C}^{∗}$. We will describe for four infinite families: two for $Sp(V)$ and two for $SO(V)$ how to combinatorially enumerate the $G^{\iota}$-orbits in $Lie2(G)$ and for each $G^{\iota}$-orbit: the Jordan decomposition of its elements, its dimension, the number of $G^{\iota}$-equivariant simple local systems (up to isomorphism) on the orbit and how to associate symbols to each of these $G^{\iota}$-equivariant simple local systems. The example 5.3 in the article “Graded Lie Algebras and Intersection Cohomology” of George Lusztig was the inspiration for our approach.

We discuss character sheaves in the setting of cyclically graded Lie algebras, focusing on cuspidal ones. Via a nearby cycle construction representations of Hecke algebras of complex reflection groups at roots of unity enter the picture. We explain a conjectural level-rank duality arising from connections with Lusztig-Yun’s work, where Fourier transforms of character sheaves are related to representations of trigonometric double affine Hecke algebras. We will also discuss connections with d-Harish-Chandra series introduced by Broué-Malle-Michel and Oblomkov-Yun’s construction of rational Cherednik algebra modules using affine Springer fibres. This is based on joint work with various co-authors, Grinberg, Liu, Trinh, Tsai, and Vilonen.

In the 1990's Lusztig generalised the classical theory of total positivity dating back to the early 20th century to arbitrary reductive algebraic groups, and beyond that to homogeneous spaces -- extending it significantly even in the GL_n setting. As part of this work, he related total positivity to the deep positivity properties of his canonical basis for quantized enveloping algebras, and he combined total positivity with tropicalisation and Langlands duality to describe the combinatorics of the canonical basis. This talk will be about the subsequent impact of this work in the context of homogeneous spaces and mirror symmetry, and vice versa, about ideas from mirror symmetry leading to new theorems on total positivity.