When
$F$
is a
$p$
-adic field, and
$G\,=\,\mathbb{G}\left( F \right)$
is the group of
$F$
-rational points of a connected algebraic
$F$
-group, the complex vector space
$\mathcal{H}\left( G \right)$
of compactly supported locally constant distributions on
$G$
has a natural convolution product that makes it into a
$\mathbb{C}$
-algebra (without an identity) called the Hecke algebra. The Hecke algebra is a partial analogue for
$p$
-adic groups of the enveloping algebra of a Lie group. However,
$\mathcal{H}\left( G \right)$
has drawbacks such as the lack of an identity element, and the process
$G\,\mapsto \,\mathcal{H}\left( G \right)$
is not a functor. Bernstein introduced an enlargement
${{\mathcal{H}}^{\hat{\ }}}\left( G \right)$
of
$\mathcal{H}\left( G \right)$
. The algebra
${{\mathcal{H}}^{\hat{\ }}}\left( G \right)$
consists of the distributions that are left essentially compact. We show that the process
$G\,\mapsto \,{{\mathcal{H}}^{\hat{\ }}}\left( G \right)$
is a functor. If
$\tau \,:\,G\,\to \,H$
is a morphism of
$p$
-adic groups, let
$F\left( \tau \right):\,{{\mathcal{H}}^{\hat{\ }}}\left( G \right)\,\to \,{{\mathcal{H}}^{\hat{\ }}}\left( H \right)$
be the morphism of
$\mathbb{C}$
-algebras. We identify the kernel of
$F\left( \tau \right)$
in terms of
$\text{Ker}\left( \tau \right)$
. In the setting of
$p$
-adic Lie algebras, with
$\mathfrak{g}$
a reductive Lie algebra,
$\mathfrak{m}$
a Levi, and
$\tau \,:\,\mathfrak{g}\,\to \,\mathfrak{m}$
the natural projection, we show that
$F\left( \tau \right)$
maps
$G$
-invariant distributions on
$\mathcal{G}$
to
${{N}_{G}}\left( \mathfrak{m} \right)$
-invariant distributions on
$\mathfrak{m}$
. Finally, we exhibit a natural family of
$G$
-invariant essentially compact distributions on
$\mathfrak{g}$
associated with a
$G$
-invariant non-degenerate symmetric bilinear form on
$\mathfrak{g}$
and in the case of
$SL\left( 2 \right)$
show how certain members of the family can be moved to the group.