Intrinsicness of Hodge-theoretic properties of Galois representations in a general reductive group

Question

In the paper "The conjectural connections between automorphic representations and Galois representations" by Buzzard and Gee, it is said

"We say that $\rho$ is crystalline/de Rham/Hodge–Tate if for some (and hence any) faithful representation $H \rightarrow GL_N$ over $\mathbb{Q}_p$ , the resulting $N$-dimensional Galois representation is crystalline/de Rham/Hodge–Tate."

Here $\rho$ is a homomorphism from the absolute Galois group of a finite extension of $\mathbb{Q}_p$ to a reductive group $H$ (defined over some algebraic closure of $\mathbb{Q}_p$). How to prove the "hence any" claim?

Second question: given that the definition does not depend on this choice anyway, is there a way to give a definition not involving this choice (or is there a natural choice here)?

Answer 1

If $H$ is reductive, any irreducible representation of $H$ is a summand of a tensor product of copies of any particular faithful representation and its dual. So it suffices to know these Hodge-theoretic properties are preserved by tensor product and summands. I think these facts can be checked explicitly from the period ring.

To see this claim about representations of reductive group, we take an arbitrary irreducible representation, view its matrix coefficients as functions on the group, thus as polynomials in the coordinates of the embedding into $GL_n$ given by the fixed faithful representation. Whatever the degree of these polynomials is tells you which tensor power you need to take to see your chosen irreducible representation as a subquotient. Then the reductive hypothesis makes it a summand.