Let M be a compact connected complex manifold and G a connected reductive complex affine algebraic group. Let $$E_G$$ E G be a holomorphic principal G–bundle over M and $$T\, \subset \, G$$ T ⊂ G a torus containing the connected component of the center of G. Let N (respectively, C) be the normalizer (respectively, centralizer) of T in G, and let W be the Weyl group N / C for T. We prove that there is a natural bijective correspondence between the following two: (1)
Torus subbundles $${\mathbb {T}}$$ T of $${\mathrm{Ad}}(E_G)$$ Ad ( E G ) such that for some (hence every) $$x\, \in \, M$$ x ∈ M , the fiber $${{\mathbb {T}}}_x$$ T x lies in the conjugacy class of tori in $${\mathrm{Ad}}(E_G)$$ Ad ( E G ) determined by T.
(2)
Quadruples of the form $$(E_W,\, \phi ,\, E'_C,\, \tau )$$ ( E W , ϕ , E C ′ , τ ) , where $$\phi \, :\, E_W\, \longrightarrow \, M$$ ϕ : E W ⟶ M is a principal W–bundle, $$\phi ^*E_G\, \supset \, E'_C\, {\mathop {\longrightarrow }\limits ^{\psi }}\, E_W$$ ϕ ∗ E G ⊃ E C ′ ⟶ ψ E W is a holomorphic reduction in structure group of $$\phi ^* E_G$$ ϕ ∗ E G to C, and $$\begin{aligned} \tau \,:\, E'_C\times N \, \longrightarrow \, E'_C \end{aligned}$$ τ : E C ′ × N ⟶ E C ′ is a holomorphic action of N on $$E'_C$$ E C ′ extending the natural action of C on $$E'_C$$ E C ′ , such that the composition $$\psi \circ \tau $$ ψ ∘ τ coincides with the composition of the quotient map $$E'_C\times N\, \longrightarrow \, (E'_C/C)\times (N/C)\,=\, (E'_C\times N)/(C\times C)$$ E C ′ × N ⟶ ( E C ′ / C ) × ( N / C ) = ( E C ′ × N ) / ( C × C ) with the natural map $$(E'_C/C)\times (N/C)\, \longrightarrow \, E_W$$ ( E C ′ / C ) × ( N / C ) ⟶ E W .
The composition of maps $$E'_C\, {\mathop {\longrightarrow }\limits ^{\psi }}\, E_W \, {\mathop {\longrightarrow }\limits ^{\phi }}\, M$$ E C ′ ⟶ ψ E W ⟶ ϕ M defines a principal
N–bundle on
M. This principal
N–bundle $$E_N$$ E N is a reduction in structure group of $$E_G$$ E G to
N. Given a complex connection $$\nabla $$ ∇ on $$E_G$$ E G , we give a necessary and sufficient condition for $$\nabla $$ ∇ to be induced by a connection on $$E_N$$ E N . This criterion relates Hermitian–Einstein connections on $$E_G$$ E G and $$E'_C$$ E C ′ in a very precise manner.