What was the role of Proserpina in Shakespeare's The Winters Tale?

Proserpina is not a character in the play but rather the
Greek goddess Persephone, who is referred to in Latin and pre-20th century English as
"Proserpina" or "Proserpine". She is mentioned by Perdita in one
passage:

I would I had some flowers o' the spring that
might
Become your time of day; and yours, and yours,
That wear upon
your virgin branches yet
Your maidenheads growing: O Proserpina,
For
the flowers now, that frighted thou let'st fall
From Dis's waggon!
daffodils,
That come before the swallow dares, and take
The winds of
March with beauty; violets dim,
But sweeter than the lids of Juno's eyes
...

The major literary sources for the story of Persephone
are the Homeric Hymn to Demeter and the Rape ("seizing" -- the sexual meaning is modern)
of Proserpine by Claudianus which was probably Shakespeare's
source.

Persephone. while picking flowers, was seized by
Hades (also called Pluto or Dis) and dragged down to the underworld. Each year as a
result of a bargain between Demerter, goddess of the harvest, and Zeus, she spends half
the year above ground (summer, when the crops grow) and half below (when Demeter morns
and nothing grows).