German ge- ptcpl cognates?

[ECOLING at aol.com]

I was just recently contemplating the augment *e-
which precedes certain completed-action verbs in
Indic and in Greek, and which, I gather from one
correspondent, is usually taken as an inheritance from
some common stage, one of several manifestations
of a close Greek-Vedic relationship.

If one takes that point of view, then it implies a Greek-Indic
common innovation compared with PIE.
Is that branch on a tree supportable?  (differs from UPenn, right?)

If not, then must one take it as an inheritance from PIE,
lost elsewhere?

it occurs to me to wonder about German

ge-  of past participles,
which (with loss of /g-/) shows up also in the "e" of the English form
"enough", related to German "genug",
from o-grade of a verb *nek- 'to reach, attain'.

What is the origin of that prefix in German?

Is it just barely conceivable that it might be related to
the Sanskrit and Greek augment,
and that it began with a laryngeal?
(I am fully aware that affixes do not always follow
the same sound changes as do roots - but also very
hesitant to posit an otherwise unproven irregularity
of sound change, so would want some pretty good
demonstrations of cognate functions, at least ones
which could develop from a common origin.
I think the functions of German <ge-> and of the
Greek and Sanskrit augment, in completive contexts,
are highly similar.

Could the origin have been something like this?
(or with a different vowel, reduced to /e/ as for
many other German unstressed verbal prefixes,
then generalized ...?)

*He-

??

Pardon, I am not a Germanicist and have
no immediate access to something that would tell me.
Pokorny's Comparative Germanic Grammar
pp.205-206 states a relation to Latin co(m)-,
but such a hypothesis is to me much more improbable on
semantic grounds.  In this view, I would assume,
the /gV-/ prefixes gradually spread from their point of
origin at the expense of other prefixes.  Perfectly possible.
Was that hypothesis posited long ago for simply for
lack of anything better, or because <ge-> shows up only in some
of the western Germanic languages (OHG, English, etc.)?
Or is there substantial support behind it,
such as details of the gradual stages of infiltration from Latin?

Best wishes,
Lloyd Anderson
Ecological Linguistics