I don't understand your definition. If $a_1,\ldots,a_n\in G$, what does $x^{a_1}\cdots x^{a_n}$, $x\in G$ mean?

No, I was mistaken. I think one can probably show that the tensor product of a Verma module with an infinite dimensional module is not finitely generated.

@Thomas $V$ is not closed under addition. It is just a set.

Have you looked at sl_2 yet? I think that will give a negative answer to your question.

That is a good point.

I believe $H(u)=u-(2u\cdot v^T)v$ is what you meant. This is a reflection. The product of two reflections is a rotation.

@YemonChoi Frobenius reciprocity seems to be okay. The proof I know is fine for the situation. The only issue might be the equality between $\dim H_\Gamma$ and the hom space, but given we have an invariant inner product, that seems fine too.

@YemonChoi I think it's okay. Isn't Fronenius reciprocity really a statement about adjointness of tensor product and hom. The proof I know is for modules over some rings S<R, though I don't recall the exact hypotheses. I will certainly double check.

As Jim Humphreys suggests, it would be helpful to know what kind of "generic irreducible representations" $\sigma$ you are considering? Do they have any special properties? The fact that you have information about the semi-simplicity of $\rho\otimes\sigma$ leads me to think there is more relevant information available.

Ahhh! I have been using the diagonal action of $C_p$ on $L$ to make computations.

I am having a bit of trouble verifying your answer. In particular, I don't see why $L$ has no invariant subspaces. For example, when $p=3$, I calculated that $L$ has exactly two 1-dimensional submodules. This means there should be a permutation covering of rank $5$. Am I missing something here?

Also, can you define you ordering more explicitly. Do you read words left-to-right or right-to-left?

I don't think your comment above is helpful. I think it was appropriate to delete it.