A depth-zero principal-series block whose Hecke algebra has a non-trivial two-cocycle
Jeffrey D. Adler, Jessica Fintzen, Kazuma Ohara
TL;DR
The paper investigates Hecke algebras arising from types for tamely ramified $p$-adic groups, focusing on depth-zero principal-series blocks. It uses an explicit $G=\mathrm{SL}_8$ example to compare the Kim–Yu-type Hecke algebra with the depth-zero counterpart, showing an isomorphism $\mathcal{H}(G^{0}(F),(K^{0},\rho^{0})) \cong \mathcal{H}(G(F),(K,\rho))$ while exhibiting a non-trivial $2$-cocycle $\mu^{\mathcal{T}}$ attached to the intertwining data. This provides a concrete instance of a depth-zero Hecke algebra that requires a non-trivial twisting, contradicting expectations that cocycles vanish in depth-zero principal-series blocks. The results refine the understanding of Kim–Yu reductions and reveal rich cocycle phenomena in affine Hecke algebras associated with depth-zero types.
Abstract
Recently the authors have shown that every Hecke algebra associated to a type constructed by Kim and Yu is isomorphic to a Hecke algebra for a depth-zero type. An example in the literature has been suggested as a counterexample to this result. We show that the example is not a counterexample, and exhibit some of its interesting properties, e.g., we show that a principal series, depth-zero type can have a Hecke algebra with non-trivial two-cocyle, a phenomenon that many did not expect could occur.