Lecture notes for Introduction to Representation Theory

Because λ is also a root of unity, the last term of the expression evaluates to
(2 + λ q−1 (ψ) + λ 1−q (ψ)).
Let’s evaluate the last summand.
Since F × is cyclic and λ q = ⇒ λ,
q 2
λ elliptic
λ q−1 (ψ) = λ 1−q (ψ) = 0.
Therefore,
λF ×
q 2 λF ×
q 2
(λ q−1 (ψ) + λ 1−q (ψ)) = − (λ q−1 (ψ) + λ 1−q (ψ)) = −2(q − 1) =
λ elliptic λF ×
q
since F × is cyclic of order q − 1. Therefore,
q
1 ⎩
2 2
ν, ν = (q − 1)
2 (q −1) (q −1)
2
1+(q −1) 1 (q −1)+ q(q − 1)
◦ · · · · ·(2(q
−q)−2(q −1)) = 1.
q(q + 1) 2
as
We have now shown that for any λ with λ q = ⇒ λ the representation Y ξ with the same character
W 1 V ,1 − V ,1 − Ind G
K C ξ
exists and is irreducible. These characters are distinct for distinct pairs (ϕ, λ) (up to switch
λ ⊃ λ q ), so there are q(q−1) such representations, each of dimension q − 1.
2
We have thus found q − 1 1-dimensional representations of G, q(q−1) 2
principal series representations,
and q(q−1) complementary series representations, for a total of q2
2
− 1 representations,
i.e., the number of conjugacy classes in G. This implies that we have in fact found all irreducible
representations of GL 2 (F q ).
4.25 Artin’s theorem
Theorem 4.73. Let X be a conjugation-invariant system of subgroups of a finite group G. Then
two conditions are equivalent:
(i) Any element of G belongs to a subgroup H X.
(ii) The character of any irreducible representation of G belongs to the Q-span of characters of
induced representations Ind G H V , where H X and V is an irreducible representation of H.
Remark. Statement (ii) of Theorem 4.73 is equivalent to the same statement with Q-span
replaced by C-span. Indeed, consider the matrix whose columns consist of the coefficients of the
decomposition of Ind G H V (for various H, V ) with respect to the irreducible representations of G.
Then both statements are equivalent to the condition that the rows of this matrix are linearly
independent.
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