A description of the integral depth-$r$ Bernstein center

In this paper we give a description of the depth-$r$ Bernstein center for non-negative integers $r$ of a reductive simply connected group $G$ over a non-archimedean local field as a limit of depth-$r$ standard parahoric Hecke algebras. Using the description, we construct maps from the algebra of stable functions on the $r$-th Moy-Prasad filtration quotient of hyperspecial parahorics to the depth-$r$ Bernstein center and use them to attach to each depth-$r$ irreducible representation $π$ an invariant $θ(π)$, called the depth-$r$ Deligne-Lusztig parameter of $π$. We show that $θ(π)$ is equal to the semi-simple part of minimal $K$-types of $π$.

💡 Research Summary

The paper studies the Bernstein center of a reductive simply‑connected group G over a non‑archimedean local field k, focusing on the depth‑r component Z_r(G) for non‑negative integer depths r. The authors first recall the Bernstein–Moy–Prasad decomposition of the category of smooth complex representations R(G) into subcategories of depth ≤ r and depth > r, which induces a corresponding decomposition of the Bernstein center Z(G)=Z_{\le r}(G)⊕Z_{>r}(G). They denote the former summand by Z_r(G) and aim to give an explicit description.

The main construction uses standard parahoric subgroups P containing a fixed Iwahori subgroup I. For each P and integer r they define the depth‑r standard parahoric Hecke algebra \