Dr. Minh Tam Trinh, Yale University

Affine Springer Fibers and Level-Rank Duality

In representation theory, affinization is the passage from a Lie-theoretic structure to its loop analogue. Oblomkov–Yun constructed (degenerate) doubly-affine Hecke actions on the modified cohomology of special affine Springer fibers, commuting with their monodromy. Ting Xue and I propose, and give evidence toward, a formula for the resulting bimodules, involving the work of Deligne–Lusztig and Broué–Malle on reductive groups over finite fields. The key is that finite fields and the field of formal complex Laurent series both have procyclic Galois group. If time permits, I will explain how our formula leads to a very general reciprocity conjecture for representations of finite reductive groups, generalizing the level-rank duality studied by Frenkel, Uglov, Chuang–Miyachi, Losev, Rouquier–Shan–Varagnolo–Vasserot, Webster...

Dr. Griffin Wang, IAS

Tetrahedral Symbol and Relative Langlands Duality

In the quantum theory of angular momentum, the Racah--Wigner coefficient, often known as the 6j symbol, is a numerical invariant assigned to a tetrahedron with half-integer edge-lengths. The 6 edge-lengths may be viewed as representations of SU(2) satisfying certain multiplicity-one conditions. One important property of the 6j symbol is its hidden symmetry outside the tetrahedral group, originally discovered by Regge. 

In this talk, we explore a generalized construction, dubbed tetrahedral symbol, in the context of rank-1 semisimple groups over local fields, and explain how the extra symmetries may be explained by relative Langlands duality. Joint work with Akshay Venkatesh.

No seminar

Prof. Boixeda Alvarez, Pablo Northeastern University

Kazhdan-Laumon category O and microlocal sheaves on affine Springer fibers

Kazhdan and Laumon introduced a category constructed by glueing W copies of the basic affine space. This category was studied by Bezrukavnikov, Polishchuk and Morton-Ferguson. In particular some sub-category was related to the representation theory of the small quantum group uq. In ongoing joint work with Morton-Ferguson we describe the Kazhdan-Laumon category as a category of microlocal sheaves. This localization is motivated by the relation between the small quantum group and microlocal sheaves on the affine Springer fiber constructed in ongoing joint work with Bezrukavnikov, McBreen and Yun. It should be interpreted as understanding some subquotient category of uq−mod constructed by some stratification of the affine Springer fiber. Understanding the filtration induced on the category by this stratification should have applications to the representation theory of uq in particular to its center and to the joint work with Bezrukavnikov, Shan and Vasserot. In this talk I will discuss the connection between the Kazhdan-Laumon category on the microlocal sheaves on affine Springer and this connection to the small quantum group.

Prof. Pramod Achar   Louisiana State University

The modular semi-infinite intersection cohomology sheaf

In 2017, Gaitsgory introduced an object called the "semi-infinite intersection cohomology sheaf" on the affine Grassmannian of a reductive group G.  I will explain what this object is and (some of) what it is good for, including zastava schemes and the Drinfeld compactification of Bun_B.  I'll then explain how to generalize this construction to coefficients in a field of positive characteristic.  Remarkably, these objects turn out to be "independent of characteristic" as long as the characteristic is good.  This is joint work with G. Dhillon and S. Riche.

No seminar

Prof. Victor Ginzburg University of Chicago

Pointwise purity, derived Satake, and Coulomb branches

Let T be a split torus and X a projective T-variety with finitely many T-fixed points. Given a Bialynicki-Birula stratification of X we consider an exact subcategory Pure^!(X)  of the T-equivariant constructible mixed derived category of X whose objects have pure and geometrically constant !-restrictions to the strata. We upgrade the equivariant cohomology functor to an exact monoidal functor from Pure^!(X) to a certain category of graded H^*_T(X)-modules  with `dual Verma flag'. Monoidal structures on the categories are given by !-tensor product. We apply the above to Iwahori equivariant sheaves on the affine Grassmannian of a reductive group G. This allows to obtain an explicit description of  the image of !-pure objects of the derived Satake category under Satake equivalence. Symplectic duality associates to any finite dimensional symplectic representation M of G (satisfying an anomaly cancellation condition)  a !-pure ring object of the Satake category whose Satake image A_M  is a commutative algebra. We prove that this algebra is finitely generated and that the scheme Spec(A_M), called the S-dual of M, is the affine closure of a normal Poisson G^\vee-variety obtained by a simple geometric construction from the Coulomb branch of M.

Dr. Kevin Lin University of Chicago

Arthur parameters via categorical trace

Recently, D. Kazhdan and A. Okounkov have obtained a new integral formula for the L^2 inner product of certain Eisenstein series which witnesses their spectral decomposition over the space of Arthur parameters. I will explain in the function field setting how the Arthur filtration is expected to be encoded geometrically, and how this point of view allows one to obtain the aforementioned result by a trace of Frobenius procedure. This is work in progress with W. Reeves.

Dr. Xinchun Ma University of Chicago

Rational Cherednik algebras and torus knots

Dr. Zeyu Wang, MIT

Higher Rankin-Selberg integrals over function fields

Yun and Zhang introduced a “higher Gross-Zagier formula” relating intersection numbers of Heeger-Drinfeld cycles on the moduli of PGL_2-shtukas to higher derivative of L-functions. Since special cycles on moduli of shtukas are rarely compact in higher-dimensional case, a generalization of the formula to higher-dimensional case was not known for a long time. In this talk, I will introduce a definition of “higher periods integrals” and a formula relating the higher periods integrals to higher derivatives of L-functions in the Rankin-Selberg case (G=GL_n * GL_{n-1}). These higher periods integrals are constructed from the RTF-algebra action of the period sheaf introduced by Ben-Zvi, Sakellaridis, and Venkatesh, and are closely related to special cycles on the moduli of shtukas. The talk will be based on my joint work with Shurui Liu (arXiv#2504.00275).

Dr. Katia Bogdanova, Harvard University

Non-vanishing of quantum geometric Whittaker coefficients

We will discuss the functor of geometric Whittaker coefficients in the context of quantum geometric Langlands program. Concretely, we will prove (modulo the spectral decomposition conjecture) that the functor of quantum geometric Whittaker coefficients is conservative on the category of cuspidal automorphic D-modules. The proof will combine generalizations of representation-theoretic and microlocal methods from the preceding works of Faergeman-Raskin and Nadler-Taylor respectively.

Prof. Thomas J. Haines, University of Maryland

Local models and nearby cycles for $\Gamma_1(p)$-level

The theory of local models has been a very successful tool for the study of Shimura varieties with parahoric level structure, and the theory is now very developed in that setting. For level structure which is deeper than Iwahori level, many complications arise, and the subject is in its infancy. I will first review the basic theory of local models for Iwahori level, concentrating on the general linear and general symplectic group cases. The main goal will be to explain what can be said about local models when the level structure is $\Gamma_1(p)$, which is slightly deeper than Iwahori level.  For PEL Shimura varieties of Siegel type, I will define the local models using a linear algebra incarnation of Oort-Tate generators of finite flat group schemes of order $p$, and then I will explain how one uses a variant of Beilinson-Drinfeld Grassmannians and Gaitsgory's central functor adapted to pro-p Iwahori level, to study the nearby cycles on the special fibers. This is based on joint work in progress with Qihang Li and Benoit Stroh.

Prof. Wei Zhang, MIT

Faltings heights and the sub-leading terms of adjoint L-functions

The Kronecker limit formula is an equality relating the Faltings height of an CM elliptic curve to the sub-leading term (at s=0) of the Dirichlet L-function of an imaginary quadratic character. Colmez conjectured a generalization relating the Faltings height of any CM abelian variety to the subleading terms of certain Artin L-functions.  In this talk we will formulate a “non-Artinian” generalization of (averaged) Colmez conjecture, relating the following two quantities:

(1) the Faltings height of certain arithmetic Chow cycles on unitary Shimura varieties, and

(2) the sub-leading term of the adjoint L-functions of (cohomological) automorphic representations of unitary groups U(n).

The $n=1$ case of our conjecture recovers the averaged Colmez conjecture.  We are able to prove our conjecture when $n=2$ using  a relative trace formula approach that can be formulated for the general $n$.

The “arithmetic relative Langlands” morally suggests that there should be a lot of other similar (at least conjectural) phenomena and we will mention some of them, including an on-going work with Tony Feng and Zhiwei Yun on the (arithmetic) volume of Shimura varieties and moduli stacks of Drinfeld Shtukas.

Joint work with Ryan Chen and Weixiao Lu.

Prof. Ivan Losev, Yale University

Harish-Chandra center for affine Kac-Moody algebras in positive characteristic

This talk is based on a joint work in progress with Gurbir Dhillon. A remarkable theorem of Feigin and E. Frenkel from the early 90's describes the center of the universal enveloping algebra of an (untwisted) affine Kac-Moody Lie algebra at the so called critical level proving a conjecture of Drinfeld: the center in question is the algebra of polynomial functions on an infinite dimensional affine space known as the space of opers. In our work we study a part of the center in positive characteristic p at an arbitrary non-critical level. Namely, we prove that the loop group invariants in the completed universal enveloping algebra is still the algebra of polynomials on an infinite dimensional affine space that is ``p times smaller than the Feigin-Frenkel center''. In my talk I will introduce all necessary notions, state the result, explain motivations and examples.    

Dr. Chun-Hsien Hsu, University of Chicago

On Poisson Summation Formulae

Braverman and Kazhdan proposed the existence of Schwartz spaces, Fourier transforms, and Poisson summation formulae on certain reductive monoids, which will allow one to prove the meromorphic continuation and functional equations of L-functions in great generality. The conjecture is refined by Lafforgue, Ngô, and is broadened to the framework of affine spherical varieties by Sakellaridis. In this talk, we will go through examples where this conjecture is established, and discuss the relations between the structure of Schwartz spaces and the Poisson summation formulae.

Prof. Tsao-Hsien Chen, University of Minnesota

Real groups, symmetric varieties and  Langlands duality

I will explain a connection between geometric Langlands on real forms of the projective line (i.e. the real projective line or the twistor P1) and relative Langlands duality, then explain recent results using this to answer some questions in Langlands duality for real groups and symmetric varieties. If time permits, I will discuss a (conjectural) connection of the story above to quantum groups.  This talk is based on joint works with Mark Macerato, David Nadler, John O'Brien, and Lingfei Yi.

Dr. Jianqiao Xia, University of Chicago

Categorical equivalences in Local Geometric Langlands Program

Let F be an equal characteristic local field and G a reductive group. It is known that by type theory that representations of G(F) can be understood as modules over various associative algebras, called Hecke algebras. In particular tamely ramified principle series are modules of the affine Hecke algebra. Based on this interpretation and a realization of the affine Hecke algebra using equivariant K-theory, Kazhdan and Lusztig proved the Deligne-Langlands correspondence. A geometrization of such correspondence was studied by Roman Bezrukavnikov, by proving a coherent realization of the affine Hecke category. In this talk, I will give an equivalence between certain deeper level Hecke categories with affine Hecke categories of a smaller group H (usually a twisted Levi of loop group LG), therefore relating wildly ramified representations with tamely ramified ones of H.

Dr. Tom Gannon, UCLA

Quantization of the universal centralizer and central D-modules

We will discuss work, joint with Victor Ginzburg, on certain quantizations (or non-commutative deformations) of objects and morphisms of interest in the geometric Langlands program. First, we will discuss the quantization of a morphism of group schemes used by Ngô in his proof of the fundamental lemma, which confirms a conjecture of Nadler. Afterwards, we will discuss how this morphism can be used to construct a D-module analogue of a recent equivalence proved by Bezrukavnikov-Deshpande in the \ell-adic setting, identifying a certain braided monoidal subcategory of the category of G-equivariant D-modules on G known as vanishing D-modules with the category of a modules for a ring known as the spherical nil-DAHA. We will also explain the construction of a certain bimodule isomorphism used to construct this braided monoidal equivalence, whose existence was originally conjectured by Ben-Zvi-Gunningham. Time permitting, we will also discuss another application of our methods: a proof of a conjecture of Braverman and Kazhdan, known as the exactness conjecture, in the D-module setting.

Dr. Justin Campbell, University of Chicago

Symmetries of geometric Eisenstein series and semi-infinite flags

I will discuss some recent work with Andreas Hayash in which we show that deformations of reducible local systems give rise to operators on the corresponding Hecke eigensheaves, obtained via the operation of geometric Eisenstein series. In the case of the trivial local system, we recover the action of the Langlands dual Lie algebra on the global intersection cohomology of quasimaps constructed by Feigin, Finkelberg, Kuznetsov, and Mirković in their 1997 paper on semi-infinite flag varieties. Our approach is philosophically similar to theirs, but we make use of some new ingredients like Gaitsgory's semi-infinite intersection cohomology sheaf.

Prof. Zhiwei Yun, MIT

Counting indecomposable G-bundles

The notion of absolutely indecomposable vector bundles over a curve naturally appear in the consideration of automorphic forms. Since they do not form an algebraic stack, it is a surprising fact that its point-counting over a finite field is still "motivic", i.e., equal to the number of points in some other stack, namely the moduli of stable Higgs bundles. This was proved by Schiffmann about 10 years ago when the degree of the vector bundle is coprime to the rank, and extended by Dobrovolsk, Ginzburg and Travkin to all degrees.

One can formulate the same counting problem for absolutely indecomposable G-bundles, where G is any connected reductive group. The previous arguments for vector bundles all used specifics about GL(n) and don't obviously generalize to G-bundles for other reductive groups G.

In joint work with Konstantin Jakob, we solve the counting problem for absolutely indecomposable G-bundles for all reductive G uniformly. We show that the number of such can be expressed using the number of stable (parabolic) G-Higgs bundles on the same curve. 

Happy thanksgiving

Prof. Dima Arinkin, University of Wisconsin

Derived category of the stack of Higgs bundles

The moduli of Higgs bundles is equipped with a Hitchin map; this lets us consider its smooth fibers are essntially abelian varieties, and singular fibers can be viewed as degenerate abelian varieties. By a results of Donagi and Pantev, the Hitchin fibrations for Langlands dual groups are generically dual to each other: the bases of the two Hitchin fibrations can be identified, and smooth fibers are dual abelian varieties. It is conjectured that the duality extends to singular fibers; such extension could be though of as the `classical limit' of the geometric Langlands correspondence.

In this talk, I will explore the derived category of coherent sheaves on the stack of Higgs bundles. Working with a stack, rather than the moduli space of semistable Higgs bundles, is natural from the point of view of the Langlands program. As we will see, this also leads to certain issues (once we leave the elliptic locus on the Hitchin base). My goal is to show that, even in the case of Higgs bundles over GL(n), the most `naive' version of the statement is inconsistent, and to see what claims have a chance of being true.