We study the deformation theory of holomorphic principal bundles over compact Riemann surfaces through symmetry analysis. Since the base curve has complex dimension one, the cohomological obstruction group \(H^2(X, {{\,\textrm{Ad}\,}}(P))\) vanishes for any coherent sheaf, so stable principal bundles are unobstructed and the moduli space is smooth at every stable point. We establish explicit dimensional formulas for the deformation spaces: for semisimple groups, \(\dim H^1(X, {{\,\textrm{Ad}\,}}(P)) = \dim (\mathfrak {g})(g-1)\) , depending only on the Lie algebra and the genus of the curve. We prove that automorphisms of the base curve induce natural linear representations on cohomology spaces preserving the cup product structure, and establish descent theorems relating equivariant deformations to cohomology on quotient surfaces. For bundles admitting reductions to maximal tori, we characterize deformations preserving these reductions and compute their dimensions, deriving closed-form formulas for all classical groups \({{\,\textrm{SL}\,}}(n,\mathbb {C})\) , \(\textrm{Sp}(2n,\mathbb {C})\) , and \(\textrm{SO}(n,\mathbb {C})\) . We further prove that no non-zero deformation of a torus reduction is invariant under the full Weyl group for any classical simple group. As applications to hyperelliptic curves, we identify the \(\tau \) -invariant deformation space with a parabolic cohomology group via the Mehta–Seshadri correspondence, and prove that all deformations of a \(\tau \) -equivariant torus reduction are \(\tau \) -anti-invariant, forming a space of dimension g.
