From pro-$p$ Iwahori–Hecke Modules to $(\varphi,\Gamma)$-Modules, II

Abstract: Let

m a t h f r a k o

be the ring of integers in a finite extension field of

{m a t h b b Q}_{p}

, let

k

be its residue field. Let

G

be a split reductive group over

{m a t h b b Q}_{p}

, let

m a t h c a l H (G, I_{0})

be its pro-

p

-Iwahori Hecke

m a t h f r a k o

-algebra. In cite{dfun} we introduced a general principle how to assign to a certain additionally chosen datum

an exact functor

from finite length

m a t h c a l H (G, I_{0})

-modules to

(v a r p h i^{r}, G a m m a)

-modules. In the present paper we concretely work out such data

for the classical matrix groups. We show that the corresponding functor identifies the set of (standard) supersingular

m a t h c a l H (G, I_{0}) o t i m e s_{m a t h f r a k o} k

-modules with the set of

(v a r p h i^{r}, G a m m a)

-modules satisfying a certain symmetry condition.