What are the easiest examples of irreducible, but not big, monodromy representations

Question

Let $\rho: \pi_1(S,s_0) \to GL(V)$ be the monodromy representation associated to a local system of $\mathbb Q$-modules $\mathbb V$ with $\mathbb V_{s_0} = V$.

Let $H$ be the Zariski closure of the image of $\rho$ in $GL(V)$. We say that $\rho$ is big if the identity component $H^0$ of $H$ acts irreducibly on $V$.

Clearly, if $\rho$ is big, then $H$ acts irreducibly on $V$.

There should be a vast amount of monodromy representations coming from geometry (e.g., coverings of Riemann surfaces, families of abelian varieties, Lefschetz pencils) so that $H$ acts irreducibly on $V$, but $H^0$ doesn't. I'm looking for explicit and natural examples.

Question. Which monodromy representations are irreducible, but not big?

Answer 1

I have seen "big monodromy" used before, in some papers of Katz I think, with a somewhat different meaning (basically that $H^0$ should as big as possible). But I'll use your definition, since that's what you seem to be interested in. One can certainly construct many non big irreducible representations as follows: Start with a variety $X$ on which a finite group $G$ in such a way that $V= H^i(X,\mathbb{Q})$ is irreducible as a $\mathbb{Q}[G]$ or $\overline{\mathbb{Q}}[G]$-module (you didn't specify which). Let $Y$ be a variety on which $G$ acts freely, then $(X\times Y)/G\to Y/G$ is big in your sense, although isotrivial. For a specific example, let $G=S_3$ acting on a Fermat cubic in $\mathbb{P}^2$, and $Y=\mathbb{A}^3$ minus the big diagonal.

Answer 2

A sage says:

What he calls “big” others call “Lie irreducible”. Simplest examples are when the monodromy is finite, e.g., a family of fibre dimension zero. Higher dimensional fibres: take a one-parameter family of, say, curves,$ f:X \rightarrow C$ over a curve $C$ as base, maybe this family has full $Sp(2g)$ as its monodromy. Now suppose $C$ has an involution, $\sigma,$ such that the family pulled back by $\sigma$ has “nothing to do” with the original family: for example, suppose the input family has nontrivial unipotent local monodromy at some point $P$ of $C,$ but that the pulled back family has trivial monodromy at this point $P.$ Then the direct sum of the two local systems will have group $Sp(2g)\times Sp(2g).$ OK, now let $D$ be the quotient of $C$ by the involution, and consider the family over D we get as the composition of $f:X \rightarrow C$ followed by the projection of $C$ to $D.$ [In terms of local systems, we have one on $C,$ then form the induced representation.] When we pull THIS family back to $C,$ we get the sum of the original local system and its $\sigma$ pullback, but the monodromy of $C/D$ switches the two factors.

In other words, the family over D has monodromy group the semidirect product of $Sp(2g)\times Sp(2g)$ with the $Z/2Z$ which switches the factors, an irred. subgroup of $Sp(4g)$, but its identity component is the reducible “diagonal” subgroup $Sp(2g)\times Sp(2g) of Sp(4g).$ Concrete example: family $y^2=x(x-1)(x-\lambda)$ over the $\lambda$ line (legendre family), involution $\lambda \mapsto -\lambda.$ Pullback family is $y^2=x(x-1)(x+\lambda).$ Input family had unip. mono. at $0$ and $1,$ pullback has unip. mono. at $0$ and $-1.$