The most common and most likely answer to this crossword clue is the 7 letter word AEOLIAN
I've seen this crossword clue in Financial Times
A
E
O
L
I
A
N
Buddy explains!

The straight part
We know that "Greek" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case greek = aeolian.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does AEOLIAN mean?
AEOLIAN noun- A lightweight fabric with a ribbed surface, generally made by combining silk and cotton, or silk and worsted warp and weft.
- Of, or relating to the wind.
- Carried, deposited or eroded by the wind.
- Aeolian mode