arXiv2026
Using distinguished pseudo-spherical representations of covering tori constructed in a companion work and Dat's parahoric induction, we construct a distinguished family of representations for an Iwahori subgroup $\bar{I}$ in $\bar{G}$, an $n$-fold Brylinski-Deligne (BD) cover of a split connected reductive $p$-adic group $G$, under explicit hypotheses on the BD datum. No tameness assumption $\mathrm{gcd}(p,n)=1$ is imposed. The construction suggests a conjectural theory of Iwahori-types for BD covers beyond the tame case. We verify this for $2$-fold covers of almost simple simply-laced simply-connected Chevalley groups of rank at least $2$ over $2$-adic fields. In particular, for these non-tame $2$-fold covers we prove: 1) The pair $(\bar{I},σ)$ is a Bushnell-Kutzko type for the Bernstein block analogous to the block of unramified principal series in a linear reductive group. This is a non-tame analogue of the Borel-Casselman theorem. 2) The Hecke algebra $\mathcal{H}(\bar{G},\bar{I},σ^{\vee})$ has an Iwahori-Matsumoto presentation up to normalization. In types $A_{r},D_{2r+1},E_{6},E_{7}$ we obtain the complete presentation and show that $\mathcal{H}(\bar{G},\bar{I},σ^{\vee})$ is isomorphic to the Iwahori-Hecke algebra of an explicit linear group. This determines an equivalence of Bernstein blocks producing a local Shimura correspondence. 3) The representation $σ$ extends to a hyperspecial maximal compact subgroup $\bar{K}$. Among the genuine $\bar{K}$-representations, $σ$ has minimal depth with respect to the principal congruence filtration. It also yields a finite Shimura correspondence, generalizing a result of Savin.