Past Algebra Seminar

Let K be an algebraically closed field. Given three elements a Lie algebra over K, we say that these elements form an sl_2-triple if they generate a subalgebra which is a homomorphic image of sl_2(K). In characteristic 0, the Jacobson-Morozov theorem provides a bijection between the orbits of nilpotent elements of the Lie algebra and the orbits of sl_2-triples. In this talk I will discuss the progress made in extending this result to fields of characteristic p, and discuss results for both the classical and exceptional Lie algebras. 

Beginning with the foundational work of Daniel Quillen, an understanding of aspects of the cohomology of finite groups evolved into a study of representations of finite groups using geometric methods of support theory. Over decades, this approach expanded to the study of representations of a vast array of finite dimensional Hopf algebras. I will discuss how related geometric and categorical techniques can be applied to linear algebra groups.

Let G and H be real reductive groups. To any L-homomorphism e: H^L \to G^L one can associate a map e_* from virtual representations of H to virtual representations of G. This map was predicted by Langlands and defined (in the real case) by Adams, Barbasch, and Vogan. Without further restrictions on e, this map can be very poorly behaved. A special case in which e_* exhibits especially nice behavior is the case when H is an endoscopic group. In this talk, I will introduce a more general class of L-homomorphisms that exhibit similar behavior to the endoscopic case. I will explain how this more general notion of endoscopic lifting relates to the theory of cohomological induction. I will also explain how this generalized notion of endoscopic lifting can be used to prove the unitarity of many Arthur packets. This is based on joint work with Jeffrey Adams and David Vogan.

Symplectic duality predicts that affine symplectic singularities come in pairs that are in a sense dual to each other. The Hikita conjecture relates the cohomology of the symplectic resolution on one side to the functions on the fixed points on the dual side.  

In a recent work with Ivan Losev and Lucas Mason-Brown, we suggested an important example of symplectic dual pairs. Namely, a Slodowy slice to a nilpotent orbit should be dual to an affinization of a certain cover of a special orbit for the Langlands dual group. In that paper, we explain that the appearance of the special unipotent central character can be seen as a manifestation of a slight generalization of the Hikita conjecture for this pair.

However, a further study shows that several things can (and do!) go wrong with the conjecture. In this talk, I will explain a modified version of the statement, recent progress towards the proof, and how special unipotent characters appear in the picture. It is based on a work in progress with Do Kien Hoang and Vasily Krylov.

The pro-C completion of a free profinite group on an infinite set of generators is a profinite group of a greater rank. However, it is still not known whether it is a free profinite group too.  We will discuss this question, present a positive answer for some special varieties, and show partial results regarding the general case. In addition, we present the infinite tower of profinite completions, which leads to a generalisation for completions of higher orders. 

The Satake isomorphism is a fundamental result in p-adic groups, and the affine Grassmannian is the natural setting where this geometrizes to the Geometric Satake Correspondence. In fact, it suffices to work with affine Grassmannian slices, which retain all of the information.

Recently, Braverman, Finkelberg, and Nakajima showed that affine Grassmannian slices arise as Coulomb branches of certain quiver gauge theories. Remarkably, their construction works in Kac-Moody type as well. Their work opens the door to studying affine Grassmannians and Geometric Satake Correspondence for Kac-Moody groups. Unfortunately, it is difficult at present to do any explicit geometry with the Coulomb branch definition. For example, a basic feature is that affine Grassmannian slices embed into one another. However, this is not apparent from the Coulomb branch definition. In this talk, I will explain why these embeddings are necessarily subtle. Nonetheless, I will show a way to construct the embeddings using fundamental monopole operators.

This is joint work with Alex Weekes.

In this lecture, I will discuss the resolution of the Arthur-Barbasch-Vogan conjecture on the unitarity of special unipotent representations for any real form of a connected reductive complex Lie group, with contributions by several groups of authors (Barbasch-Ma-Sun-Zhu, Adams-Arancibia-Mezo, and Adams-Miller-van Leeuwen-Vogan). The main part of the lecture will be on the approach of the first group of authors for the case of real classical groups: counting by coherent families (combinatorial aspect), construction by theta lifting (analytic aspect), and distinguishing by invariants (algebraic-geometric aspect), resulting in a full classification, and with unitarity as a direct consequence of the construction.

The plethysm product on symmetric functions corresponds to composition of polynomial representations of general linear groups. Decomposing a plethysm product into Schur functions, or equivalently, writing the corresponding composition of Schur functors as a direct sum of Schur functors, is one of the main open problems in algebraic combinatorics. I will give an introduction to these mathematical objects emphasising the beautiful interplay between representation theory and combinatorics. I will end with new results obtained in joint work with Rowena Paget (University of Kent) on stability on plethysm coefficients. No specialist background knowledge will be assumed.

Rational Cherednik algebra is a flat deformation of a skew product of the Weyl algebra and a Coxeter group W. I am going to discuss two interesting subalgebras of Cherednik algebras going back to the work of Hakobyan and the speaker from 2015. They are flat deformations of skew products of quotients of the universal enveloping algebras of gl_n and so_n, respectively, with W. They also have to do with particular nilpotent orbits and generalised Howe duality.  Their central quotients can be given as the algebra of global sections of sheaves of Cherednik algebras. The talk is partly based on a joint work with D. Thompson.

Let F be a p-adic local field and let G be a connected split reductive group over F. Let H be the pro-p Iwahori-Hecke algebra of the p-adic group G(F), with coefficients in an algebraically closed field k of characteristic p. The module theory over H (or a certain derived version thereof) is of considerable interest in the so-called mod p local Langlands program for G(F), whose aim is to relate the smooth modular representation theory of G(F) to modular representations of the absolute Galois group of F. In this talk, we explain a possible construction of a certain moduli space for those Galois representations into the Langlands dual group of G over k which are semisimple. We then relate this space to the geometry of H. This is work in progress with Cédric Pépin.

I will explain an ongoing program, joint with Kari Vilonen, that aims to study unitary representations of real reductive Lie groups using mixed Hodge modules on flag varieties. The program revolves around a conjecture of Schmid and Vilonen that natural filtrations coming from the geometry of flag varieties control the signatures of Hermitian forms on real group representations. This conjecture is expected to facilitate new progress on the decades-old problem of determining the set of unitary irreducible representations by placing it in a more conceptual context. Our results to date centre around the interaction of Hodge theory with the unitarity algorithm of Adams, van Leeuwen, Trapa, and Vogan, which calculates the signature of a canonical Hermitian form on an arbitrary representation by reducing to the case of tempered representations using deformations and wall crossing. Our results include a Hodge-theoretic proof of the ALTV wall crossing formula as a consequence of a more refined result and a verification of the Schmid-Vilonen conjecture for tempered representations.

Kazhdan-Lusztig polynomials are remarkable polynomials associated to pairs of elements in a Coxeter group W. They describe the base change between the standard and Kazhdan-Lusztig bases for the corresponding Hecke algebra. They were discovered by Kazhdan and Lusztig in 1979 and have found applications throughout representation theory and geometry. In 1987, Deodhar introduced the parabolic Kazhdan-Lusztig polynomials associated to a Coxeter group W and a standard parabolic subgroup P. These describe the base change between the standard and Kazhdan-Lusztig bases for the anti-spherical module for the Hecke algebra. (We recover the original definition of Kazhdan and Lusztig by taking the trivial parabolic subgroup).

(Anti-spherical) Hecke categories first rose to mathematical celebrity as the centrepiece of the proof of the (parabolic) Kazhdan-Lusztig positivity conjecture. The Hecke category categorifies the Hecke algebra and the anti-spherical Hecke category categorifies the anti-spherical module. More precisely, it was shown by Elias-Williamson (and Libedinsky-Williamson) that the (parabolic) Kazhdan-Lusztig polynomials are precisely the graded decomposition numbers for the (anti-spherical) Hecke categories over fields of characteristic zero, hence proving positivity of their coefficients.
The (anti-spherical) Hecke categories can be defined over any field. Their graded decomposition numbers over fields of positive characteristic p, the so-called (parabolic) p-Kazhdan-Lusztig polynomials, have been shown to have deep connections with the modular representation theory of reductive groups and symmetric groups. However, these polynomials are notoriously difficult to compute.
Unlike in the case of the ordinary (parabolic) Kazhdan-Lusztig polynomials, there is not even a recursive algorithm to compute them in general.
In this talk, I will discuss the representation of the anti-spherical Hecke categories for (W,P) a Hermitian symmetric pair, over an arbitrary field. In particular, I will explain why the decomposition numbers are characteristic free in this case.
This is joint work with C. Bowman, A. Hazi and E. Norton.

Let F be a p-adic local field and let G be a connected split reductive group over F. Let H be the pro-p Iwahori-Hecke algebra of the p-adic group G(F), with coefficients in an algebraically closed field k of characteristic p. The module theory over H (or a certain derived version thereof) is of considerable interest in the so-called mod p local Langlands program for G(F), whose aim is to relate the smooth modular representation theory of G(F) to modular representations of the absolute Galois group of F. In this talk, we explain a possible construction of a certain moduli space for those Galois representations into the Langlands dual group of G over k which are semisimple. We then relate this space to the geometry of H. This is a work in progress with Cédric Pépin.

Perverse sheaves are an indispensable tool in representation theory. Their stalks often encode important representation theoretic information such as composition multiplicities or canonical bases. For the nilpotent cone, there is an algorithm that computes these stalks, known as the Lusztig-Shoji algorithm. In this talk, we discuss how this algorithm can be modified to compute stalks of perverse sheaves on more general varieties. As an application, we obtain a new algorithm for computing canonical bases in certain quantum groups as well as composition multiplicities for standard modules of the affine Hecke algebra of $\mathrm{GL}_n$.

(jt work with David Cushing and George Stagg)

Prolog is a rather unusual programming language that was developed by Alain Colmerauer 50 years ago in one of the buildings on the way to the CIRM in Luminy. It is a declarative language that operates on a paradigm of first-order logic -- as distinct from imperative languages like C, GAP and Magma. Prolog operates by loading in a list of axioms as input, and then responds at the command line to queries that ask the language to achieve particular goals, given those axioms. It gained some notoriety through IBM’s implementation of ‘Watson’, which was a system designed to play the game show Jeopardy. Through a very efficiently implemented constraint logic programming module, it is also the worlds fastest sudoku solver. However, it has had barely any serious employment by pure mathematicians. So the aim of this talk is to advertise Prolog through an extended example: my co-authors and I used it to search for new simple Lie algebras over the field GF(2) and were able to classify a certain flavour of absolutely simple Lie algebra in dimensions 15 and 31, discovering a dozen or so new examples. With some further examples in dimension 63, we then extrapolated two previously undocumented infinite families of simple Lie algebras.

I will explain how bornological and condensed structures can both be described as algebraic theories. I will also show how this permits the construction of functors between bornological and condensed structures. If time permits I will also briefly describe how to compare condensed derived geometry and bornological derived geometry and sketch how they relate to analytic geometry and Arakelov geometry

The local Langlands conjectures connect representations of p-adic groups to certain representations of Galois groups of local fields called Langlands parameters.  Recently, there has been a shift towards studying representations over more general coefficient rings and towards certain categorical enhancements of the original conjectures.  In this talk, we will focus on representations over coefficient rings with p invertible and how the corresponding category of representations of the p-adic group decomposes.  

In the last few years, major advances have been made in our understanding of the representation theory of reductive algebraic groups over algebraically closed fields of positive characteristic. Four key tools which are central to this progress are highest weight theory, reduction to the principal block, wall-crossing functors, and tilting modules. When considering instead the representation theory of the Lie algebras of these algebraic groups, more subtleties arise. If we look at those modules whose p-character is in so-called standard Levi form we are able to recover the four tools mentioned above, but they have been less well-studied in this setting. In this talk, we will explore the similarities and differences which arise when employing these tools for the Lie algebras rather than the algebraic groups. This research is funded by a research fellowship from the Royal Commission for the Exhibition of 1851.

We discuss the Local Langlands correspondence in connection with the Bernstein center and the Stable Bernstein center. We also give an example of stable Bernstein center as a stable essentially compact invariant distribution.

Fibres coming from the Springer resolution on the nilpotent cone are incredibly rich algebraic varieties that have many applications in representation theory and combinatorics. Though their geometry can be very difficult to describe in general, in type A at least, their irreducible components can be described using standard Young tableaux, and this can help describe their geometry in small dimensions. In this talk, I will report on recent and ongoing work with Lewis Topley and separately Daniele Rosso on geometrical and combinatorial applications of the classical ‘type A’ Springer fibres and the ‘exotic’ type C Springer fibres coming from Kato’s exotic Springer correspondence.

Khovanov-Rozansky homology is a link invariant that categorifies the HOMFLY-PT polynomial. I will describe a geometric model for this invariant, living in the monodromic Hecke category. I will also explain how it allows to identify objects representing graded pieces of Khovanov-Rozansky homology, using a remarkable family of character sheaves. Based on joint works with Roman Bezrukavnikov.

In this talk we will present some work in progress generalising Lubin-Tate formal group laws to higher dimensions. There have been some other generalisations, but ours is different in that the ring over which the formal group law is defined changes as the dimension increases. We will state some conjectures about these formal group laws, including their relationship to the Drinfeld tower over the p-adic upper half plane, and provide supporting evidence for these conjectures.

We outline Dirac cohomology for Lie algebras and Vogan’s conjecture. We then cover some basic material on Schur-Weyl duality and Arakawa-Suzuki functors. Finishing with current efforts and results on generalising Vogan’s conjecture to a Schur-Weyl duality setting. This would relate the centre of a Lie algebra with the centre of the relevant tantaliser algebra. We finish by considering a unitary module X and giving a bound on the action of the tantalizer algebra.

A finite W-algebra is a gadget associated to each nilpotent orbit in a complex semisimple Lie algebra g. There is a functor from W-modules to a full subcategory of g-modules, known as Skryabin’s equivalence, and every primitive ideals of the enveloping algebra U(g) as the annihilator of a module obtained in this way. This gives a convenient way of organising together primitive ideals in terms of nilpotent orbits, and this approach has led to a resurgence of interest in some hard open problems which lay dormant for some 20 years. The primitive ideals of U(g) which come from one-dimensional representations of W-algebras are especially nice, and we shall call them Losev—Premet ideals. The goal of this talk is to explain my recent work which seeks to: (1) describe the structure of the space of the dimensional representations of a finite W-algebra and (2) classify the Losev—Premet ideals.

I will discuss algebraic conditions that for a given group guarantee or characterize the vanishing of cohomology in a given degree with coefficients in any unitary representation. These conditions will be expressed in terms positivity of certain elements over group algebras, where positivity is meant as being a sum of hermitian squares. I will explain how conditions like this can be used to give computer-assisted proofs of vanishing of cohomology.