We give an introductory overview of Coulomb branches and how they relate to Hecke algebras. For example, one concrete application is the construction of canonical basis for (spherical) double affine Hecke algebras. The goal is to explore how to take further advantage of this point of view.

We will survey some examples of where $sl(2)$ acts on structures appearing in categorification including link homology. This is joint work with You Qi, Louis-Hadrien Robert, and Emmanuel Wagner.

We propose an approach to categorification of the colored Jones polynomial evaluated at a $2p$-th root of unity. This is done by by equipping a $p$-differential discovered by Cautis on the triply graded Khovanov-Rozansky homology, defined in terms of (singular) Soergel bimodules. This is based on joint work with Louis-Hadrien Robert, Joshua Sussan and Emmanuel Wagner.

Given a $p$-adic reductive group $G$ and a field $k,$ we are interested in the category $\mathcal{C}$ of smooth representations of $G$ over $k$.

When $k=\mathbb{C}$, this category is fairly well understood. An important tool that comes into play when studying its blocks is (generalizations of) the Hecke algebra $H_{\mathrm{\Omega }}$ attached to an open compact subgroup $\mathrm{\Omega }.$ For example, if $\mathrm{\Omega }$ is an Iwahori subgroup, then it is easy to see that $H_{\mathrm{\Omega }}$ is an affine Hecke algebra with usual braid and quadratic relations (with nonzero parameter).

When $k$ has characteristic $p$, the category $\mathcal{C}$ is poorly understood. In this case, it makes sense to focus on the Hecke algebra corresponding to the pro-$p$ Sylow subgroup ${\mathrm{\Omega }}_p$ of an Iwahori subgroup. The resulting Hecke algebra is now (almost) a Nil Hecke algebra which does tell us something about $\mathcal{C},$ but in fact it is a DG version of it that is more relevant in this context.

We will talk about the role played by this DG algebra and its cohomology algebra in understanding the (derived) category of smooth representations of $G$ over $k.$

The uniqueness of Whittaker models plays an important role in the representation theory of linear reductive groups due to its relation to L-functions. However, such uniqueness fails in general for nonlinear covering groups. We investigate this failure of uniqueness through the Gelfand-Graev representation, which is the dual of the Whittaker space.

Using the pro-$p$ Iwahori-Hecke algebra, we describe the Iwahori-fixed vectors in the Gelfand-Graev representation of covering groups as a module over the Iwahori-Hecke algebra, generalizing work of Barbasch-Moy and Chan-Savin for linear groups. As applications, we 1) relate the Gelfand-Graev representation to the metaplectic representation of Sahi-Stokman-Venkateswaran; 2) compute the dimension of the space of Whittaker models for constituents of certain unramified principal series; 3) make a connection to quantum algebra through quantum affine Schur-Weyl duality in type $A$.

This is joint work with Fan Gao and Nadya Gurevich.

We describe the space of Whittaker functions on an $n$-fold cover of a $p$-adic group $G$ as a module over the spherical Hecke algebras of $G.$ To do so, we first review some affine Hecke algebra combinatorics introdued by Lascoux-Leclerc-Thibon (in type A) and Haiman-Grojnowski (in general) which turns out to be relevant to this question, and also links the answer to quantum groups at an $n$-th root of unity. This is joint work with Valentin Buciumas.

The classical Satake transform gives an isomorphism between the complex spherical Hecke algebra of a $p$-adic reductive group $G$, and the Weyl-invariants of the complex spherical Hecke algebra of a maximal torus of $G$. This provides a way for understanding the $K$-invariant vectors in smooth irreducible complex representations of $G$ (where $K$ is a maximal compact subgroup of $G$, and allows one to construct instances of unramified Langlands correspondences. In this talk, I'll present work in progress with Cédric Pépin in which we attempt to understand the analogous situation with $\mathrm{mod}p$ coefficients, and working at the level of the derived category of smooth $G$-representations.