Let
be a prime number,
a nonarchimedean local
field with residue field
of characteristic
,
and
an algebraically closed field of characteristic different from
. We investigate the irreducible
smooth
-representations
of
. The components of an
irreducible smooth
-representation
of
restricted
to
form an
-packet
. We use the classification
of such
to determine
the cardinality of
,
which is
or
.
When
we have to use the Langlands correspondence for
. When
is a prime number
distinct from
and
, we determine the behaviour
of an integral
-packet under
reduction modulo
. We prove a
Langlands correspondence for
,
and an enhanced one when the characteristic
of is not
. Finally,
pursuing a theme of Henniart and Vignéras (2024), which studied the case of inner forms
of
,
we show that near identity a nontrivial irreducible smooth
-representation
of
is, up to a finite-dimensional representation, isomorphic to a sum of
or
representations
in an
-packet
of size
(when
is odd there is only
one such
-packet).
We show that for
in an
-packet of size
and a sufficiently large
integer
, the dimension of
the invariants of
by the
-th congruence subgroup
of an Iwahori or a pro-
Iwahori subgroup of
is equal to
,
with
if
is odd
and
,
otherwise
is an integer. We also study the fixed points by the
-th
congruence subgroups of the maximal compact subgroups of
where the answer
depends on the parity of
.
Keywords
modular irreducible representations, $L$-packets, Whittaker
spaces, local Langlands correspondence