# Invariant polynomials on $\mathfrak{gl}(r, \mathbb{R})$ and characteristic classes

I'm currently studying from the book of Loring Tu, 'Differential Geometry: Connections, Curvature and Characteristic classes', chapter 23.

This chapter is about the Chern-Weil homomorphism $$c_E : \text{Inv} (\mathfrak{gl}(r, \mathbb{R})) \rightarrow H^{*} (M)$$ from the algebra $$\text{Inv}(\mathfrak{gl}(r, \mathbb{R}))$$ of invariant polynomials on $$\mathfrak{gl}(r, \mathbb{R})$$ to the de Rham cohomology algebra of $$M$$.

The way that these invariant polynomials are defined is: Let $$X = [x^{i}_j]$$ be an $$r \times r$$ matrix with indeterminate entries $$x^{i}_{j}$$. A polynomial $$P(X)$$ on $$\mathfrak{gl}(r, \mathbb{R})$$ is a polynomial in the entries of $$X$$. A polynomial $$P(X)$$ on $$\mathfrak{gl}(r, \mathbb{R})$$ is said to be $$GL(r, \mathbb{R})$$-invariant or simply invariant if $$P(A^{-1} X A) = P(X)$$ for all $$A \in GL(r, \mathbb{R})$$.

The author gives $$\text{tr}(X)$$ and $$\det(X)$$ as examples of invariant polynomials on $$\mathfrak{gl}(r, \mathbb{R})$$.

My question is: is there any specific reason we consider polynomials on the Lie algebra $$\mathfrak{gl}(r, \mathbb{R})$$? Why not for example on the general linear group $$GL(r, \mathbb{R})$$? And does this have some generalizations to arbitrary Lie algebras?

Keep in mind what you are going to use these invariant polynomials for: you are going to feed them the curvature 2-form of your connection in order to produce a real-valued $$2k$$-form. The curvature is a Lie algebra-valued 2-form. Why is it valued in the Lie algebra instead of the group?
An intuitive way to think about the connection as a Lie algebra valued 1-form $$\omega$$ is "if I parallel transport a vector an infinitesimally small amount in the tangent direction $$X$$, then with nearby tangent spaces identified I am rotating my vector by the infinitesimal amount $$\omega(X)$$." So since "infinitesimal rotations" live in the Lie algebra and not the Lie group, it is necessary that the connection be Lie algebra-valued. Then its curvature also ends up Lie algebra-valued, and so the polynomials that eat the curvature form must be polynomials on the Lie algebra.
As for generalizations, I don't trust myself to say anything about "arbitrary Lie algebras," but certainly when $$\mathfrak{g}$$ is the Lie algebra of a Lie group $$G$$ the theory has a generalization (and maybe some restriction is necessary on $$G$$, I wouldn't know). The setting for that would be Chern-Weil theory applied to principal $$G$$-bundles. A good reference is Dupont, Fibre Bundles and Chern-Weil Theory.