Lecture notes for Introduction to Representation Theory

⇒
4.24.3 Principal series representations
Let
(the set of upper triangular matrices); then
∼
B → G, B = {
0
∼ }
∼
and
|B| = (q − 1) 2 q,
1 ∼
[B, B] = U = { },
0 1
B/[B, B] ∪ = F ×
q × F× q
(the isomorphism maps an element of B to its two diagonal entries).
Let
∂ : B ⊃ C ×
be a homomorphism defined by
a b
∂ = ∂ 1 (a)∂ 2 (c),for some pair of homomorphisms ∂ 1 , ∂ 2 : F × q ⊃ C × .
0 c
Define
V 1 , 2
= Ind G
B C ,
where C is the 1-dimensional representation of B in which B acts by ∂. We have
dim(V 1 , 2
) = |G| = q + 1.
|B|
Theorem 4.71.
1. ∂ 1 = ⇒ ∂ 2 ≥ V 1 , 2
is irreducible.
2. ∂ 1 = ∂ 2 = µ ≥ V 1 , 2
= C µ W µ , where W µ is a q-dimensional irreducible representation of
G.
3. W ∪
∪
1 , ⊗ {∂ ⊗ ⊗
µ = W ξ if and only if µ = λ; V 1 , 2
= V ⊗ if and only if {∂
2
1 , ∂ 2 } = 1 , ∂ 2 } (in the
⊗ ⊗
second case, ∂ 1 = ∂ 2 , ∂ = ∂ 2 ).
Proof. From the Mackey formula, we have
1 ⇒
1
tr V1 , 2
(g) =
∂(aga −1 ).
|B|
aG, aga −1 B
If
the expression on the right evaluates to
x 0
g = ,
0 x
∂(g) |G| = ∂ 1 (x)∂ 2 (x) ⎩ q + 1 .
|B|
If
x 1
g = ,
0 x
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