Lower bounds on the essential dimension of reductive groups

We introduce a technique for proving lower bounds on the essential dimension of split reductive groups. As an application, we strengthen the best previously known lower bounds for various split simple algebraic groups, most notably for the exceptional group $E_8$. In the case of the projective linear group ${\mathrm{PGL}}_n$, we recover A. Merkurjev’s celebrated lower bound with a simplified proof. Our technique relies on decompositions of loop torsors over valued fields due to P. Gille and A. Pianzola.