Existence of types for all Bernstein blocks

Establish that every Bernstein block Rep(G)_{[L, ρ]} admits a Bushnell–Kutzko type (K, π), i.e., a pair consisting of a compact open subgroup K ⊂ G and an irreducible representation π of K such that ind_K^G π lies in Rep(G)_{[L, ρ]} and is a compact generator of that block.

Background

Types provide compact generators for Bernstein blocks that are compatible with compact induction, enabling a link between block-wise K-theory and subgroup data. While types are known for GL_n and, via related work, for SL_n, the general existence across all Bernstein blocks remains conjectural.

The authors summarize the state of knowledge, pointing to Bushnell–Kutzko’s and Roche–Goldberg’s results for specific groups and Fintzen’s recent advances, but emphasize the conjectural status in full generality.