Forthcoming events in this series


Tue, 20 Feb 2024

14:00 - 15:00
L5

Faithfulness of highest-weight modules for Iwasawa algebras

Stephen Mann
(University of Cambridge)
Abstract

Iwasawa algebras are completions of group algebras for p-adic Lie groups, and have applications for studying the representations of these groups. It is an ongoing project to study the prime ideals, and more generally the two-sided ideals, of these algebras.

In the case of Iwasawa algebras corresponding to a simple Lie algebra with a Chevalley basis, we aim to prove that all non-zero two-sided ideals have finite codimension. To prove this, it is sufficient to show faithfulness of modules arising from highest-weight modules for the corresponding Lie algebra.

I have proved two main results in this direction: firstly, I proved the faithfulness of generalised Verma modules over the Iwasawa algebra. Secondly, I proved the faithfulness of all infinite-dimensional highest-weight modules in the case where the Lie algebra has type A. In this talk, I will outline the methods I used to prove these cases.

Tue, 13 Feb 2024

14:00 - 15:00
L5

Functional Calculus, Bornological Algebra, and Analytic Geometry

Jack Kelly
((University of Oxford))
Abstract

Porta and Yue Yu's model of derived analytic geometry takes as its category of basic, or affine, objects the category opposite to simplicial algebras over the entire functional calculus Lawvere theory. This is analogous to Lurie's approach to derived algebraic geometry where the Lawvere theory is the one governing simplicial commutative rings, and Spivak's derived smooth geometry, using the Lawvere theory of C-infinity-rings. Although there have been numerous important applications including GAGA, base-change, and Riemann-Hilbert theorems, these methods are still missing some crucial ingredients. For example, they do not naturally beget a good definition of quasi-coherent sheaves satisfying descent. On the other hand, the Toen-Vezzosi-Deligne approach of geometry relative to a symmetric monoidal category naturally provides a definition of a category of quasi-coherent sheaves, and in two such approaches to analytic geometry using the categories of bornological and condensed abelian groups respectively, these categories do satisfy descent.  In this talk I will explain how to compare the Porta and Yue Yu model of derived analytic geometry with the bornological one. More generally we give conditions on a Lawvere theory such that its simplicial algebras embed fully faithfully into commutative bornological algebras. Time permitting I will show how the Grothendieck topologies on both sides match up, allowing us to extend the embedding to stacks.

This is based on joint work with Oren Ben-Bassat and Kobi Kremnitzer, and follows work of Kremnitzer and Dennis Borisov.

Tue, 30 Jan 2024

14:00 - 15:00
L5

Equivariant vector bundles with connection on the p-adic half-plane

Simon Wadsley
(University of Cambridge)
Abstract

Recent joint work with Konstantin Ardakov has been devoted to classifying equivariant line bundles with flat connection on the Drinfeld p-adic half-plane defined over F, a finite extension of Q_p, and proving that their global sections yield admissible locally analytic representations of GL_2(F) of finite length. In this talk we will discuss this work and invite reflection on how it might be extended to equivariant vector bundles with connection on the p-adic half-plane and, if time permits, to higher dimensional analogues of the half-plane.

Tue, 23 Jan 2024

14:00 - 15:00
L5

On a quantitative version of Harish-Chandra's regularity theorem and singularities of representations

Yotam Hendel
(KU Leuven)
Abstract

Let G be a reductive group defined over a local field of characteristic 0 (real or p-adic). By Harish-Chandra’s regularity theorem, the character Θ_π of an irreducible representation π of G is given by a locally integrable function f_π on G. It turns out that f_π has even better integrability properties, namely, it is locally L^{1+r}-integrable for some r>0. This gives rise to a new singularity invariant of representations \e_π by considering the largest such r.

We explore \e_π, show it is bounded below only in terms of the group G, and calculate it in the case of a p-adic GL(n). To do so, we relate \e_π to the integrability of Fourier transforms of nilpotent orbital integrals appearing in the local character expansion of Θ_π. As a main technical tool, we use explicit resolutions of singularities of certain hyperplane arrangements. We obtain bounds on the multiplicities of K-types in irreducible representations of G for a p-adic G and a compact open subgroup K.

Based on a joint work with Itay Glazer and Julia Gordon.

Tue, 05 Dec 2023

14:00 - 15:00
L6

Representation type of cyclotomic quiver Hecke algebras

Qi Wang
(Tsinghua University)
Abstract

One of the fundamental problems in representation theory is determining the representation type of algebras. In this talk, we will introduce the representation type of cyclotomic quiver Hecke algebras, also known as cyclotomic Khovanov-Lauda-Rouquier algebras, especially in affine type A and affine type C. Our main result relies on novel constructions of the maximal dominant weights of integrable highest weight modules over quantum groups. This talk is based on collaborations with Susumu Ariki, Berta Hudak, and Linliang Song.

Tue, 28 Nov 2023

14:00 - 15:00
L5

Hecke algebras for p-adic groups and explicit Local Langlands Correspondence

Yujie Xu
(Columbia University (New York))
Abstract

I will talk about several results on Hecke algebras attached to Bernstein blocks of (arbitrary) reductive p-adic groups, where we construct a local Langlands correspondence for these Bernstein blocks. Our techniques draw inspirations from the foundational works of Deligne, Kazhdan and Lusztig. 

As an application, we prove the Local Langlands Conjecture for G_2, which is the first known case in literature of LLC for exceptional groups. Our correspondence satisfies an expected property on cuspidal support, which is compatible with the generalized Springer correspondence, along with a list of characterizing properties including the stabilization of character sums, formal degree property etc. In particular, we obtain (not necessarily unipotent) "mixed" L-packets containing "F-singular" supercuspidals and non-supercuspidals. Such "mixed" L-packets had been elusive up until this point and very little was known prior to our work. I will give explicit examples of such mixed L-packets in terms of Deligne-Lusztig theory and Kazhdan-Lusztig parametrization. 

If time permits, I will explain how to pin down certain choices in the construction of the correspondence using stability of L-packets; one key input is a homogeneity result due to Waldspurger and DeBacker. Moreover, I will mention how to adapt our general strategy to construct explicit LLC for other reductive groups, such as GSp(4), Sp(4), etc. Such explicit description of the L-packets has been useful in number-theoretic applications, e.g. modularity lifting questions as in the recent work of Whitmore. 

Some parts of this talk are based on my joint work with Aubert, and some other parts are based on my joint work with Suzuki. 
 

Mon, 20 Nov 2023

15:00 - 16:00
L6

t-structures on the equivariant derived category of the Steinberg scheme.

Ivan Losev
(Yale University)
Abstract

The Steinberg scheme and the equivariant coherent sheaves on it play a very important role in Geometric Representation theory. In this talk we will discuss various t-structures on the equivariant derived category of the Steinberg of importance for Representation theory in positive characteristics. Based on arXiv:2302.05782.

Tue, 14 Nov 2023

14:00 - 15:00
L5

Fourier and Small ball estimates for word maps on unitary groups

Itay Glazer
(University of Oxford )
Abstract

Let w(x_1,...,x_r) be a word in a free group. For any group G, w induces a word map w:G^r-->G. For example, the commutator word w=xyx^(-1)y^(-1) induces the commutator map. If G is finite, one can ask what is the probability that w(g_1,...,g_r)=e, for a random tuple (g_1,...,g_r) of elements in G.

In the setting of finite simple groups, Larsen and Shalev showed there exists epsilon(w)>0 (depending only on w), such that the probability that w(g_1,...,g_r)=e is smaller than |G|^(-epsilon(w)), whenever G is large enough (depending on w).

In this talk, I will discuss analogous questions for compact groups, with a focus on the family of unitary groups; For example, given r independent Haar-random n by n unitary matrices A_1,...,A_r, what is the probability that w(A_1,...,A_r) is contained in a small ball around the identity matrix?

Based on a joint work with Nir Avni and Michael Larsen.  

 

Tue, 07 Nov 2023

14:00 - 15:00
L5

A solution functor for D-cap-modules

Finn Wiersig
(University of Oxford)
Abstract

The theory of D-modules has found remarkable applications in various mathematical areas, for example, the representation theory of complex semi-simple Lie algebras. Two pivotal theorems in this field are the Beilinson-Bernstein Localisation Theorem and the Riemann-Hilbert Correspondence. This talk will explore a p-adic analogue. Ardakov-Wadsley introduced the sheaf D-cap of infinite order differential operators on a given smooth rigid-analytic variety to develop a p-adic counterpart for the Beilinson-Bernstein localisation. However, the classical approach to the Riemann-Hilbert Correspondence does not apply in the p-adic context. I will present an alternative approach, introducing a solution functor for D-cap-modules using new methods from p-adic Hodge theory.

Tue, 31 Oct 2023
14:00
L5

Elliptic representations

Dan Ciubotaru
(Oxford)
Abstract

In representation theory, the characters of induced representations are explicitly known in terms of the character of the inducing representation. This leads to the question of understanding the elliptic representation space, i.e., the space of representations modulo the properly (parabolically) induced characters. I will give an overview of the description of the elliptic space for finite Weyl groups, affine Weyl groups, affine Hecke algebras, and their connection with the geometry of the nilpotent cone of a semisimple complex Lie algebra. These results fit together in the representation theory of semisimple p-adic groups, where they lead to a new description of the elliptic space within the framework of the local Langlands parameterisation.

Tue, 24 Oct 2023

14:00 - 15:00
L5

Existence and rotatability of the two-colored Jones–Wenzl projector

Amit Hazi
(University of York)
Abstract

The two-colored Temperley-Lieb algebra is a generalization of the Temperley-Lieb algebra. The analogous two-colored Jones-Wenzl projector plays an important role in the Elias-Williamson construction of the diagrammatic Hecke category. In this talk, I will give conditions for the existence and rotatability of the two-colored Jones-Wenzl projector in terms of the invertibility and vanishing of certain two-colored quantum binomial coefficients. As a consequence, we prove that Abe’s category of Soergel bimodules is equivalent to the diagrammatic Hecke category in complete generality.

 

Tue, 17 Oct 2023
14:00
L5

Microlocal sheaves and affine Springer fibers

Pablo Boixeda Alvarez
(Yale University)
Abstract

The resolutions of Slodowy slices e are symplectic varieties that contain the Springer fiber (G/B)e as a Lagrangian subvariety. In joint work with R. Bezrukavnikov, M. McBreen, and Z. Yun, we construct analogues of these spaces for homogeneous affine Springer fibers. We further understand the categories of microlocal sheaves in these symplectic spaces supported on the affine Springer fiber as some categories of coherent sheaves.

In this talk I will mostly focus on the case of the homogeneous element ts for s a regular semisimple element and will discuss some relations of these categories with the small quantum group providing a categorification of joint work with R.Bezrukavnikov, P. Shan and E. Vasserot.

Tue, 13 Jun 2023

14:00 - 15:00
L4

Correspondences of affine Hecke algebras in the Langlands program

Anne-Marie Aubert
((Mathematics Institute of Jussieu-Paris Left Bank, Sorbonne University ))
Abstract

The irreducible smooth representations of p-adic reductive groups and the enhanced Langlands parameters of these latter can both be partitioned into series indexed by "cuspidal data". On the representation side, cuspidality refers to supercuspidal representations of Levi subgroups, while on the Galois side, it refers to "cuspidal unipotent pairs", as introduced by Lusztig, in certain subgroups of the Langlands dual groups.

In addition, on both sides, the elements in a given series are in bijection with the simple modules of a generalized affine Hecke algebra. 

The cuspidal data on one side are expected to be in bijection with the cuspidal data on the other side. We will formulate conditions on this bijection that will guarantee the existence of a bijection between the simple modules of the attached generalized affine Hecke algebras. For the exceptional group of type G_2 and for all pure inner forms of quasi-split classical groups, the Hecke algebras are actually isomorphic.

Tue, 06 Jun 2023

14:00 - 15:00
L6

The wavefront set of unipotent representations with real infinitesimal character

Emile Okada
(National University of Singapore)
Abstract

For a reductive group defined over a p-adic field, the wavefront set is an invariant of an admissible representations which roughly speaking measures the direction of the singularities of the character near the identity. Studied first by Roger Howe in the 70s, the wavefront set has important connections to Arthur packets, and has been the subject of thorough investigation in the intervening years. One of main lines of inquiry is to determine the relation between the wavefront set and the L-parameter of a representation. In this talk we present new results answering this question for unipotent representations with real infinitesimal character. The results are joint with Dan Ciubotaru and Lucas Mason-Brown.

Tue, 30 May 2023

14:00 - 15:00
L6

The Jacobson-Morozov Theorem in positive characteristic

Rachel Pengelly
(Birmingham University)
Abstract

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. 

Mon, 29 May 2023

15:30 - 16:30
L5

Modular representations theory: from finite groups to linear algebraic groups

Eric M. Friedlander
(University of Southern California)
Abstract

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.

Tue, 23 May 2023

14:00 - 15:00
L6

Endoscopic lifting and cohomological induction

Lucas Mason-Brown
Abstract

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.

Mon, 22 May 2023

16:00 - 17:00
C4

On the Hikita-Nakajima conjecture for Slodowy slices

Dmytro Matvieievskyi
(Kavli IPMU)
Abstract

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.

Tue, 16 May 2023

14:00 - 15:00
L6

Profinite completion of free profinite groups

Tamar Bar-On
(University of Oxford)
Abstract

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. 

Tue, 09 May 2023

14:00 - 15:00
L6

Fundamental monopole operators and embeddings of Kac-Moody affine Grassmannian slices

Dinakar Muthiah
(University of Glasgow)
Abstract

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.

Fri, 05 May 2023

15:00 - 16:00
L4

On the Arthur-Barbasch-Vogan conjecture

Chen-Bo Zhu
(National University of Singapore)
Abstract

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.

Tue, 02 May 2023

14:00 - 15:00
L6

An introduction to plethysm

Mark Wildon
(Royal Holloway, University of London)
Abstract

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.

Tue, 25 Apr 2023

14:00 - 15:00
L6

Subalgebras of Cherednik algebras

Misha Feigin
(University of Glasgow)
Abstract

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.

Tue, 18 Apr 2023

14:00 - 15:00
L6

Modular Hecke algebras and Galois representations

Tobias Schmidt
(University of Rennes)
Abstract

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.

Tue, 28 Mar 2023

14:00 - 15:00
C4

Mixed Hodge modules and real groups

Dougal Davis
(University of Melbourne)
Abstract

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.