Irreducible local Galois representation with arbitrary Hodge-Tate weights

Question

Let $p$ be a prime and $M$ be a finite multiset of non-negative integers. Does there exist a continuous irreducible de Rham representation $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to GL_n(\mathbb{Q}_p)$ whose Hodge-Tate weights are $M$? Can the representation be chosen to be crystalline as well?

Answer 1

Let me suppose (for simplicity) that $M$ has distinct elements.

Choose a polynomial $P \in \mathbf{Q}_p[X]$ with distinct roots, all of which have the same valuation, with this common valuation being the average of the elements of $M$. Let $A$ be a matrix with characteristic polynomial $P$.

[Edit: All valuations being the same is stronger than needed here, I guess; it should be sufficient that the Newton polygon of $P$ lie above the Hodge polygon determined by $M$, with the same endpoints, and they don't touch anywhere else.]

If we're given a full flag $F^0 \supsetneq F^1 \supsetneq \dots F^n = \{0\}$ of subspaces in $\mathbf{Q}_p^n$, then we can define a filtered $\varphi$-module $D$ by taking $\mathbf{Q}_p^n$ with $\varphi = A$ and $Fil^i D = $(whichever of the $F^i$ has the correct dimension).

If $F$ is in general position with respect to the eigenspaces of $A$, then for every $\varphi$-stable subspace $E \subseteq D$, the jumps in the filtration of $E$ are the $d$ smallest jumps in the filtration $F$. So $E$ will satisfy $t_H(E) \le t_N(E)$, with strict inequality unless $E = D$ or $E = 0$.

So the Galois representation $V$ with $D_{cris}(V^*) = D$ is irreducible with Hodge-Tate weights $M$.