Geometry ag.algebraica – Analogue of the main package for the Hodge package

Leave $ X $ Being a complex projective variety connected without problems.

A holomorphic Higgs package is a pair. $ (E, theta) $ consists of a pack of holomorphic vectors $ E $ in $ X $ together with a Higgs field $ theta in H ^ 0 (X, End (E) otimes Omega_X ^ 1) $ with $ theta wedge theta = 0 $ in $ H ^ 0 (X, End (E) otimes Omega ^ 2_X) $.

A holomorphic Higgs package. $ (E, theta) $ It is said to have a structure of a Hodge package system if $ E $ It has a holomorphic decomposition of direct sum. $ E = bigoplus limits_ {i = 0} ^ n E_i $ such that $ theta (E_i) subseteq E_ {i-1} otimes Omega_X ^ 1 $, for all $ 1 leq i leq n $.

Now let $ G $ be an algebraic cognate reducer group connected through $ mathbb {C} $.

A holomorphic director. $ G $-Higgs package in $ X $ it's a pair $ (E_G, theta) $, where $ E_G $ he is a holomorphic director $ G $-Bundle in $ X $ Y $ theta in H ^ 0 (X, text {ad} (E_G) otimes Omega_X ^ 1) $ such that $ theta wedge theta = 0 $ in $ H ^ 0 (X, text {ad} (E_G) otimes Omega_X ^ 2) $; here $ text {ad} (E_G): = E_G times ^ G mathfrak {g} $ is the attached vector pack associated with the attached representation $ ad: G longrightarrow End ( mathfrak {g}) $Y $ mathfrak {g} $ is the lie algebra of $ G $.

For example when $ G = GL_n $, $ text {ad} (E_G) = End (E_ {GL_n}) $, where $ E_ {GL_n} $ is the vector pack corresponding to the $ GL_n $-make $ E_G $.

Question:
Is there any analog of Hodge's package system for the principal case? $ G $-Higgs packages?