Lecture notes for Introduction to Representation Theory

We thus have
Ind G
K C = ∪
ξ q Ind G
K C ξ
because they have the same character. Therefore, for λ q = ⇒
Next, we look at the following tensor product:
W 1 V ,1 ,
λ we get 2
1
q(q − 1) representations.
where 1 is the trivial character and W 1 is defined as in the previous section. The character of this
representation is
x 0
ν = q(q + 1)ϕ(x);
0 x
Thus the ”virtual representation”
ν(A) = 0 for A parabolic or elliptic;
x 0
ν = ϕ(x) + ϕ(y).
0 y
W 1 V ,1 − V ,1 − Ind G
K C ξ ,
where ϕ is the restriction of λ to scalars, has character
x 0
ν = (q − 1)ϕ(x);
0 x
In all that follows, we will have λ q = ⇒ λ.
x 1
ν = −ϕ(x);
0 x
x 0
ν = 0;
0 y
x πy x πy x πy
ν = −λ − λ q .
y x y x y x
The following two lemmas will establish that the inner product of this character with itself is
equal to 1, that its value at 1 is positive. As we know from Lemma 4.27, these two properties imply
that it is the character of an irreducible representation of G.
Lemma 4.72. Let ν be the character of the ”virtual representation” defined above. Then
◦ν, ν = 1
and
ν(1) > 0.
Proof.
ν(1) = q(q + 1) − (q + 1) − q(q − 1) = q − 1 > 0.
We now compute the inner product ◦ν, ν. Since ϕ is a root of unity, this will be equal to
1
(q−1) (q−1)
2 2 q(q − 1)
1+(q−1) 1 (q −1)+
(q − 1) 2 · · · ·
·
q(q + 1)
2
74
(λ(ψ)+λ q (ψ))(λ(ψ) + λ q (ψ))
λ elliptic