In this talk we will briefly recall how quantum groups at roots give rise Verlinde algebras which can be realised as Grothendieck rings of certain monoidal categories. The ring structure is quite interesting and was very much studied in type A. I will try to explain how one gets a natural action of certain double affine Hecke algebras and show how known properties of these rings can be deduced from this action and in which sense modularity of the tensor category is encoded.
James's Conjecture predicts that the adjustment matrix for weight w blocks of the Iwahori-Hecke algebras ℋn and the q-Schur algebras 𝒮n is the identity matrix when w < char(F). Fayers has proved James's Conjecture for blocks of ℋn of weights 3 and 4. We shall discuss some results on adjustment matrices that have been used to prove James's Conjecture for blocks of 𝒮n of weights 3 and 4 in an upcoming paper. If time permits, we will look at a proof of the weight 3 case.
We define the affinization of an arbitrary monoidal category, corresponding to the category of string diagrams on the cylinder. We also give an alternative characterization in terms of adjoining dot generators to the category. The affinization formalizes and unifies many constructions appearing in the literature. We describe a large number of examples coming from Hecke-type algebras, braids, tangles, and knot invariants.
There are two categorical realizations of the affine Hecke algebra: constructible sheaves on the affine flag variety and coherent sheaves on the Langlands dual Steinberg variety. A fundamental problem in geometric representation theory is to relate these two categories by a category equivalence. This was achieved by Bezrukavnikov in characteristic 0 about a decade ago. In this talk, I will discuss a first step toward solving this problem in the modular case joint with R. Bezrukavnikov and S. Riche.
We discuss a new explicit isomorphism between (truncations of) quiver Hecke algebras and Elias-Williamson’s diagrammatic endomorphism algebras of Bott-Samelson bimodules. This allows us to deduce that the decomposition numbers of these algebras (including as examples the symmetric groups and generalised blob algebras) are tautologically equal to the associated p-Kazhdan-Lusztig polynomials, provided that the characteristic is greater than the Coxeter number. This allows us to give an elementary and explicit proof of the main theorem of Riche-Williamson’s recent monograph and extend their categorical equivalence to cyclotomic Hecke algebras, thus solving Libedinsky-Plaza’s categorical blob conjecture.
We parametrize elements in the full Hecke algebra in a way such that the parametrization represents a generic automorphic form. By convolving, we then arrive at pre-trace formulas which are modular in three variables. From here, various identities for higher moments may be derived. We give applications to the sup-norm and fourth-norm of holomorphic Hecke eigenforms as well as Hecke-Maass forms on Γ \ ℍ and furthermore outline future work on higher moments of periods and quantum variance. This is joint work with Ilya Khayutin.
The point of this talk is to give three examples of derived structures influencing representations that have connections with number theory. These structures arise from the differential graded algebra of group cochains valued in the endomorphism ring of a representation.
Two examples have to do with representations of a Galois group. One of these realizes a number theoretic criterion for the modulo p multiplicity one condition for Jacobians of modular curves at an Eisenstein maximal ideal of a Hecke algebra; this is joint work with Preston Wake. Another furnishes a realization as a derived Galois deformation ring of an exterior algebra considered in works of Galatius-Venkatesh, Hansen-Thorne, and Venkatesh. The third example features smooth modulo p representations of a p-adic Lie group, answering some questions of Sorensen about the relationship between its Iwasawa algebra and the associated derived Hecke algebra.
