| Résume | Etingof--Ginzburg introduced in 2002 the rational Cherednik algebra $H_{t,c}(G,V)$ for any finite irreducible complex reflection group $(G,V)$. This is a deformation of the algebra $\mathrm{S}(V \otimes V^*) \rtimes \mathbb{C} G$ depending on two parameters $t$ and $c$. The representation theory of the algebras $H_{0,c}$ helps to understand the geometry of the symplectic variety $(V \otimes V^*)/G$. The restricted rational Cherednik algebra $\overline{H}_c$ of $(G,V)$ in $c$ is now a particular canonical finite-dimensional quotient of $H_{0,c}$ and thus captures a certain part of the representation theory of $H_{0,c}$. Martino conjectured in 2009 a surprising relationship between the blocks of $\overline{H}_c$ for certain $c$ and Rouquier blocks of $(G,V)$. While this conjecture and a lot of further results about the restricted rational Cherednik algebras are already proven for the infinite series $G(m,p,n)$, not much is known in the case of exceptional complex reflection groups. In my talk I will present a new computational method which attempts to compute the radical of a finite-dimensional module over a finite-dimensional algebra (also in characteristic zero). I used this method to obtain new results about the restricted rational Cherednik algebras for exceptional complex reflection groups. Moreover, I will present a counter-example to the generic version of Martino's conjecture. |
