so it's been 1 month since I started reading Serge Lang Algebra, I'm now pretty advanced in my reading of the book but I am stuck at p 721, here it is :
So here Lang is computing and classifying all the irreducible complex characters of the general linear group of dimension 2 over a finite field.
The part I'm having trouble with is when he constructs the fourth type of irreducible character, specifically the claim :
"Note that a direct computation using Frobenius reciprocity shows that θᴳ occurs in the character (res θ , λ)ᴳ."
If you want to see how he defines the two characters, here it is :
So I've tried to show that θᴳ occurs in the decomposition of (res θ , λ)ᴳ into a linear combination of irreducible characters, but to me it's not at all clear how to do it because we're dealing with reducible characters.
I've been able to show that θ occurs in the linear decomposition of the restriction of (res θ , λ)ᴳ to the group associated to the character θ. I simply used the inner product (or bilinear form if you want) of the characters and the orthogonality relations of irreducible characters.
But I cannot extend my methods to prove that θᴳ is contained in (res θ , λ)ᴳ because those characters are reducible.
Any help would be appreciated.