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Let $G$ be a compact group, $H$ a normal open subgroup, and $K$ a $p$-adic field (so that not all $G$-reps with coefficients in $K$ are semisimple). Let $V$ be a topological $K$-vector space with continuous action of $H$, such that $V$ is irreducible as an $H$-representation (and therefore finite-dimensional, since $H$ is compact). Is the induced representation $\text{Ind}_H^G(V)$ semisimple?

In this specific context, I'm thinking of $G$ as the Galois group of a field, if that's helpful.

Thanks in advance!

Let $G$ be a compact group, $H$ a normal open subgroup, and $K$ a $p$-adic field (so that not all $G$-reps with coefficients in $K$ are semisimple). Let $V$ be a finite-dimensional topological $K$-vector space with continuous action of $H$, such that $V$ is irreducible as an $H$-representation. Is the induced representation $\text{Ind}_H^G(V)$ semisimple?

In this specific context, I'm thinking of $G$ as the Galois group of a field, if that's helpful.

Thanks in advance!

Let $G$ be a compact group, $H$ a normal open subgroup, and $K$ a $p$-adic field (so that not all $G$-reps with coefficients in $K$ are semisimple). Let $V$ be a topological $K$-vector space with continuous action of $H$ (and therefore finite-dimensional, since $H$ is compact). Is the induced representation $\text{Ind}_H^G(V)$ semisimple?

In this specific context, I'm thinking of $G$ as the Galois group of a field, if that's helpful.

Thanks in advance!

Let $G$ be a compact group, $H$ a normal open subgroup, and $K$ a $p$-adic field (so that not all $G$-reps with coefficients in $K$ are semisimple). Let $V$ be a topological $K$-vector space with continuous action of $H$, such that $V$ is irreducible as an $H$-representation (and therefore finite-dimensional, since $H$ is compact). Is the induced representation $\text{Ind}_H^G(V)$ semisimple?

In this specific context, I'm thinking of $G$ as the Galois group of a field, if that's helpful.

Thanks in advance!

Let $G$ be a compact group, $H$ a normal open subgroup, and $K$ a $p$-adic field (so that not all $G$-reps with coefficients in $K$ are semisimple). Let $V$ be a topological $K$-vector space with continuous action of $H$ (and therefore finite-dimensional, since $H$ is compact) with coefficients in $K$, is the induced representation $\text{Ind}_H^G(V)$ semisimple?

In this specific context, I'm thinking of $G$ as the Galois group of a field, if that's helpful.

Thanks in advance!

Let $G$ be a compact group, $H$ a normal open subgroup, and $K$ a $p$-adic field (so that not all $G$-reps with coefficients in $K$ are semisimple). Let $V$ be a topological $K$-vector space with continuous action of $H$ (and therefore finite-dimensional, since $H$ is compact). Is the induced representation $\text{Ind}_H^G(V)$ semisimple?

In this specific context, I'm thinking of $G$ as the Galois group of a field, if that's helpful.

Thanks in advance!