I will present arguably the most basic one among the set of conjectures stated in 1998 by Broue, Malle and Rouquier (following early work by Broue and Malle) about the generalized Iwahori-Hecke algebras associated to complex reflection groups. By a combination of several kind of arguments and lots of hand-writen as well as computer-assisted calculations, it seems that a complete proof is now within reach. I will report on recent progress by my PhD student E. Chavli, as well as on a recent work by G. Pfeiffer and myself on this topic.[-]

Let $p$ be a prime number and $F$ be a non-archimedean field with finite residue class field of characteristic $p$. Understanding the category of Iwahori-Hecke modules for $SL_2(F)$ is of great interest in the study of $p$-modular smooth representations of $SL_2(F)$, as these modules naturally show up as spaces of invariant vectors under the action of the standard pro-$p$-Iwahori subgroup. In this talk, we will discuss a work in progress in which we aim to classify all non-trivial extensions between these modules and to compare them with their analogues for $p$-modular smooth representations of $SL_2(F)$ and with their Galois counterpart in the setting of the local Langlands correspondences in natural characteristic.[-]

In geometric representation theory, one is interested in studying the geometry of affine Grassmannians of quasi-split simply-connected reductive groups. In this endeavor, one of the main techniques, introduced by Faltings in the split case, consists in constructing natural realisations of these ind-schemes over the integers. In the twisted case, this was done by Pappas and Rapoport in the tamely ramified case, i.e. over $\mathbb{Z}[1/e]$, where $e = 2$ or $3$ is the order of the automorphism group of the split form we are dealing with. We explain how to extend the parahoric group scheme that appeared in work of Pappas, Rapoport, Tits and Zhu to the polynomial ring $\mathbb{Z}[t]$ with integer coefficients and additionally how the group scheme obtained in char. $e$ can be regarded as a parahoric model of a basic exotic pseudo-reductive group. Then we study the geometry of the affine Grassmannian and also its global deformation à la Beilinson-Drinfeld, recovering all the known results in the literature away from $e = 0$. This also has some pertinence to the study of local models of Shimura varieties in wildly ramified cases.[-]

Superpolynomials are formed with $N$ commuting and anti-commuting (skew) variables. By considering the space of skew variables of fixed degree as a module of the symmetric group $\mathcal{S}_{N}$ the theory of generalized Jack polynomials constructed by S Griffeth can be used to define nonsymmetric Jack superpolynomials. We present the theory, give details about the structure and derive norm formulas. Denote the parameter by $\kappa$ then the norm is positive-definite for $-\frac{1}{N}<\kappa<\frac{1}{N}$. Analogously there is a structure as Hecke algebra $\mathcal{H}_{N}(t)$-module on the skew polynomials and this allows the use of the theory of vectorvalued $(q, t)$-Macdonald polynomials studied by J-G Luque and the author. We outline the theory and present norm formulas and evaluations at special points. The norm is positive-definite for $q>0$ and min $(q^{1 / N}, q^{-1 / N}) < t < max (q^{1 / N}, q^{-1 / N} )$. As in the scalar case the evaluations use $(q, t)$-hook products.[-]

I will talk about my joint work with Aubert where we prove the Local Langlands Conjecture for $G_2$ (explicitly). This uses our earlier results on Hecke algebras attached to Bernstein components of (arbitrary) reductive $p$-adic groups, as well as an expected property on cuspidal support, along with a list of characterizing properties (including stability). In particular, we obtain 'mixed' L-packets containing F-singular supercuspidals and nonsupercuspidals. Our methods are inspired by the Langlands-Shahidi method, Deligne-Lusztig and Lusztig theories etc. If time permits, I will explain how to characterize our correspondence using stability of L-packets, by computing character formulae in terms of (generalized) Green functions ; one key input is a homogeneity result due to Waldspurger and DeBacker. Moreover, I will mention how to adapt our general strategy to construct LLC for other reductive groups, such as $G S p(4), S p(4)$, etc. The latter parts are based on recent joint work with Suzuki.[-]

Using the representation theory of Cherednik algebra at t= 0, we define a family of "Calogero-Moser cellular characters" for any complex reflection group $W$. Whenever $W$ is a Coxeter group, we conjecture that they coincide with the "Kazhdan-Lusztig cellular characters". We shall give some evidences for this conjecture. Our main result is that, whenever the associated Calogero-Moser space is smooth, then all the Calogero-Moser cellular characters are irreducible. This implies in particular that our conjecture holds in type $A$ and for some particular choices of the parameters in type $B$.[-]