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