Assumption in Fulton & Harris? (representations of $SL_2(q)$)

Question

(Originally asked at https://math.stackexchange.com/questions/2399091/restricted-representations-and-irreducibility-in-sl-2q)

I'm working through the Fulton-Harris ad-hoc method of constructing the character table of $SL_2(q)$ (for $q$ an odd prime power), $\S$5.2 for reference.

I am at a point where we are considering representations of $GL_2(q)$, and how they decompose on restriction to a subgroup of index 2 (specifically the subgroup of elements with square determinant, which we will call $H$). We then consider the further restriction from this subgroup to $SL_2(q)$.

I understand almost everything. There are two irreducible representations of $GL_2(q)$, say $W,X$, that each split into a direct sum of irreducible representations of $H$: $\ \ W|_{H}=W'\oplus W''$, $X|_H = X'\oplus X''$. We know the dimensions of these representations - each is half the dimension of the larger rep.

Now, we have classified all irreducible representations of $SL_2(q)$ except 4, whose dimensions are known to be the same as $W',W'',X',X''$. The book then claims (bottom of page 72):

"This shows that these representations ($W',W'',X',X''$) stay irreducible on restriction from $H$ to $SL_2(q)$."

I'm confused as to why they make this assumption - or am I overlooking something obvious? Thanks in advance.

Answer 1

They don't assume anything. They just carry out the argument you describe. They explain that the restriction of the rep to $H$ splits as a direct sum of two irreducibles and then in the next sentence they explain that since the restriction of the rep to $SL_2(q)$ is also known to consists of two irreducibles by character theory, the irreducible pieces appearing in the restriction to $H$ must remain irreducible, when we restrict to $SL_2(q)$.