MIT Lie Groups Seminar
2022 - 2023
Meetings: 4:00pm on Wednesdays
This seminar will take place either in-person or online. For in-person seminars, it will be held at 2-142. You are welcome to join in-person seminars by Zoom. For remote participation, the Zoom link is the same as last year's. You can email Andre Dixon or Ju-Lee Kim for the Zoom meeting Link. To access videos of talks, please email Andre Dixon for the password.
Local geometric langlands and affine Beilinson-Bernstein localization
Abstract: We will exposit some tenets of the local geometric Langlands philosophy with concrete representation-theoretic consequences. In particular, we will focus on the problem of affine Beilinson-Bernstein localization at the critical level. Then, we will explain how to actually prove some of these statements. Some of this work is joint with Sam Raskin.
Wave-front set and graded Springer theory
Abstract: For characters of p-adic reductive groups there is the notion of wave-front set, which is a set of nilpotent orbits that describes the asymptotic behavior of a character near the identity. (Maybe one can think of the closure of wave-front set as singular support.) There is a long-standing conjecture that any wave-front set is contained in a single geometric orbit, as worked out by many authors for several types of depth-0 representations. In this talk, we explain how the above conjecture cannot hold in general, because an analogous assertion does not hold for graded Lie algebras. We will discuss this last statement in the context of graded Springer theory.
Wavefront sets and unipotent representations of p-adic groups
Abstract: An important invariant for admissible representations of reductive p-adic groups is the wavefront set, the collection of the maximal nilpotent orbits in the support of the orbital integrals that occur in the Harish-Chandra-Howe local character expansion. We compute the geometric and Okada's canonical unramified wavefront sets for representations in Lusztig's category of unipotent reduction for a split group in terms of the Kazhdan-Lusztig parameters. We use this calculation to give a new characterisation of the anti-tempered unipotent Arthur packets. Another interesting consequence is that the geometric wavefront set of a unipotent supercuspidal representation uniquely determines the nilpotent part of the Langlands parameter; this is an extension to p-adic groups of Lusztig's result for unipotent representations of finite groups of Lie type. The talk is based on joint work with Lucas Mason-Brown and Emile Okada.
On the formal degree conjecture for classical groups
Abstract: A conjecture of Hiraga, Ichino and Ikeda expresses the formal degree of a discrete series of a (algebraic) reductive group over a local field in terms of the adjoint gamma factor of its Langlands parameter. It can be checked for real reductive groups using work of Harish Chandra and the explicit form of the Langlands correspondence in this case. For classical groups over a p-adic field, the conjecture was established for odd orthogonal groups as well as unitary groups by two completely different methods. In this talk, I will explain a proof for symplectic and even orthogonal p-adic groups based on properties of twisted endoscopy and ideas originating from Shahidi relating residue of intertwining operators to twisted orbital integrals. This method can actually be readily adapted to treat odd orthogonal and unitary groups as well.
Affine Springer fibers and sheaves on Hilbert scheme of points on the plane.
Abstract: My talk is based on the joint work with E. Gorsky and O. Kivinen. I will explain a construction that associates a coherent sheaf on the Hilbert scheme of points on the plane to plane curve singularity. The global sections of the sheaf are equal to cohomology of the corresponding Affine (type A) Springer fiber. The construction categorifies HOMFLYPT homology/cohomogy of compactified Jacobian conjecture if combined with Soergel bimodule/ Sheaves of Hilbert scheme theorem of Oblomkov-Rozansky. I will also discuss generalizations outside of type A.
From geometric Langlands to classical via the trace of Frobenius
Abstract: I'll start by summarizing the main results of the series [AGKRRV], where it is shown that the trace of Frobenius on the category of automorphic sheaves with nilpotent singular support identifies with the space of unramified automorphic functions. We'll then discuss conjectural counterparts of this statement in the local and global ramified settings.
Multiplicative Hitchin fibration and Fundamental Lemma
Abstract: Given a reductive group G and some auxiliary data, one has the Hitchin fibration associated with the adjoint action of G on Lie(G), which is successfully used by B. C. Ngô to prove the endoscopic fundamental lemma for Lie algebras. Following the same idea, there is a group analogue called the multiplicative Hitchin fibration by replacing the Lie algebra with reductive monoids, and one can hope to directly prove the fundamental lemma at group level. This project is close to completion and we report the results so far. There are many new features that are not present in the additive case, among which is a pleasant surprise that there might be some strata in the support theorem that are not explained by endoscopy.
On some Hecke algebra modules arising from theta correspondence and it’s deformation
Abstract: This talk is based on the joint work with Jiajun Ma and Congling Qiu on theta correspondence of type I dual pairs over a finite field F_q. We study the Hecke algebra modules arising from theta correspondence between certain Harish-Chandra series for these dual pairs. We first show that the normalization of the corresponding Hecke algebra is related to the first occurrence index, which leads to a proof of the conservation relation. We then study the deformation of this Hecke algebra module at q=1 and generalize the results of Aubert-Michel-Rouquier and Pan on theta correspondence between unipotent representations along this way.
Alexander Bertoloni Meli
Comparison of local Langlands correspondences for odd unitary groups
Abstract: I will speak on joint work with Linus Hamann and Kieu Hieu Nguyen. For odd unramified unitary groups over a p-adic field, there is a Langlands correspondence using trace formula techniques due to Mok and Kaletha--Minguez--Shin--White. Another correspondence was constructed recently by Fargues--Scholze using p-adic geometry. We show these correspondences are compatible by first proving an analogous result for unitary similitude groups by studying the cohomology of local Shimura varieties. If time permits, we will discuss applications to the Kottwitz conjecture and eigensheaf conjecture of Fargues.
Moduli space of flower curves
Abstract: The Deligne-Mumford moduli space of genus 0 curves plays many roles in representation theory. For example, the fundamental group of its real locus is the cactus group which acts on tensor products of crystals. I will discuss a variant on this space which parametrizes "flower curves". The fundamental group of the real locus of this space is the virtual cactus group. This moduli space of flower curves is also the parameter space for inhomogeneous Gaudin algebras.
Perverse sheaves on symmetric products of the plane
Abstract: In joint work with Tom Braden we give a purely algebraic description of the category of perverse sheaves (with coefficients in any field) on $S^n(C^2)$, the n-fold symmetric product of the plane. In particular, using the geometry of the Hilbert scheme of points, we relate this category to the symmetric group and its representation ring. Our work is motivated by analogous structure appearing in the Springer resolution and Hilbert-Chow morphism..
Categorical q-deformed rational numbers and compactifications of stability space
Abstract: We discuss new categorical interpretations of two distinct q-deformations of the rational numbers. The first one was introduced in a different context by Morier-Genoud and Ovsienko, and enjoys fascinating combinatorial, topological, and algebraic properties. The second one is a natural partner to the first, and is new. We obtain these deformations via boundary points of a compactification of the space of Bridgeland stability conditions on the 2-Calabi--Yau category of the A2 quiver. The talk is based on joint work with Louis Becker, Anand Deopurkar, and Anthony Licata.
Webs for symplectic and orthogonal groups
Abstract: In joint work with Ben Elias, David Rose, and Logan Tatham, we give a generators and relations presentation of the monoidal category generated by fundamental representations of the symplectic group (and its q-analogue). I will discuss our work, which generalizes both the Temperely-Lieb algebra description of representations of SL_2 and Kuperberg’s ``spider" description of representations of Sp_4. I will then comment on joint work in progress with Haihan Wu which gives an analogous presentation of the monoidal category generated by exterior powers of the vector representation of the orthogonal group (and its q-analogue).
Kazhdan-Laumon Category O, Schwartz space, and the semi-infinite flag variety
Abstract: We define an analogue of Category O in the context of Kazhdan and Laumon's 1988 gluing construction for perverse sheaves on the basic affine space G/U. We explicitly classify its simple objects, and then use our understanding of the structure of this category to discuss its connections to some other interesting objects in representation theory, namely Braverman-Kazhdan's Schwartz space on G/U and perverse sheaves on the semi-infinite flag variety.
Contact: Andre Dixon