# semisimple restricted representation

As a consequence of THEOREM 2.6 in [ROBERT J. BLATTNER, SUSAN MONTGOMERY, CROSSED PRODUCTS AND GALOIS EXTENSIONS OF HOPF ALGEBRAS, PACIFIC JOURNAL OF MATHEMATICS Vol. 137, No. 1, 1989, 37-54], we know that:

Let $p$ be a prime, $H$ be a normal subgroup of $G$ with $(p,|G:H|)=1$ and $V$ be a left $kG$-module, where the characteristic of $k$ is $p$. If $V$ is a semisimple $kH$-module, then $V$ is a semisimple $kG$-module.

This consequence tells us that sometimes the complete reducibility can be deduced from the restricted representation.

Question: Besides the above consequence, are there any other similar theorems?

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It is probably obvious to you, but I think it is should be mentioned that the other way round, you have Clifford's theorem in modular representation theory of finite groups : Theorem (Clifford) If $H$ is any normal subgroup of $G$ and $V$ is a semi-simple $k[G]$-module, then $V_{|H}$ is a semi-simple $k[H]$-module. – Niels Jun 5 '11 at 8:04
Yes. But my question is that: if the restricted representation $V|_{H}$ is semisimple, in what conditions can we deduce that $V$ is semisimple? Here $H$ need not be a normal subgroup. Furthermore, we consider several restricted representations $\{V|_{H_{i}}\}$. If the restricted representations $\{V|_{H_{i}}\}$ are semisimple, in what conditions can we deduce that $V$ is semisimple? – sife Jun 5 '11 at 10:45
@sife: Your last question needs a more precise formulation concerning the indices of the subgroups relative to the given prime number. Like the original question asked, this further question seems too loose to answer directly. – Jim Humphreys Jun 5 '11 at 11:36

The statement quoted here is not due originally to Blattner and Montgomery, but is a familiar generalization of the standard "averaging" proof of Maschke's Theorem on complete reducibility for finite groups: see for example Exercise 8 in Section 10 of the book Representations of Finite Groups and Associative Algebras by Curtis and Reiner (1962). As they note, the group $G$ could be infinite.