AbstractLet X be a complex irreducible smooth projective curve, and let $${{\mathbb {L}}}$$
L
be an algebraic line bundle on X with a nonzero section $$\sigma _0$$
σ
0
. Let $${\mathcal {M}}$$
M
denote the moduli space of stable Hitchin pairs $$(E,\, \theta )$$
(
E
,
θ
)
, where E is an algebraic vector bundle on X of fixed rank r and degree $$\delta $$
δ
, and $$\theta \, \in \, H^0(X,\, {\mathcal {E}nd}(E)\otimes K_X\otimes {{\mathbb {L}}})$$
θ
∈
H
0
(
X
,
E
n
d
(
E
)
⊗
K
X
⊗
L
)
. Associating to every stable Hitchin pair its spectral data, an isomorphism of $${\mathcal {M}}$$
M
with a moduli space $${\mathcal {P}}$$
P
of stable sheaves of pure dimension one on the total space of $$K_X\otimes {{\mathbb {L}}}$$
K
X
⊗
L
is obtained. Both the moduli spaces $${\mathcal {P}}$$
P
and $${\mathcal {M}}$$
M
are equipped with algebraic Poisson structures, which are constructed using $$\sigma _0$$
σ
0
. Here we prove that the above isomorphism between $${\mathcal {P}}$$
P
and $${\mathcal {M}}$$
M
preserves the Poisson structures.
Metaplectic quantization of the moduli spaces of flat and parabolic bundles.
