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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⇒<br />
the expression evaluates <strong>to</strong><br />
since here<br />
If<br />
the expression evaluates <strong>to</strong><br />
since here<br />
∂(g) · 1,<br />
aga −1 B ≥ a B.<br />
x 0<br />
g = ,<br />
0 y<br />
⎩<br />
∂1 (x)∂ 2 (y) + ∂ 1 (y)∂ 2 (x) · 1,<br />
If<br />
aga −1 B ≥ a B or a is an element of B multiplied by the transposition matrix.<br />
x πy<br />
g = , x = y<br />
y x<br />
the expression on the right evaluates <strong>to</strong> 0 because matrices of this type don’t have eigenvalues over<br />
F q (and thus cannot be conjugated in<strong>to</strong> B). From the definition, ∂ i (x)(i = 1, 2) is a root of unity,<br />
so<br />
|G|◦ν V1 , 2<br />
, ν V1 , 2<br />
= (q + 1) 2 (q − 1) + (q 2 − 1)(q − 1)<br />
The last two summands come from the expansion<br />
2 2<br />
+ 2(q + q) (q − 1)(q − 2) + (q + q) ∂ 1 (x)∂ 2 (y)∂ 1 (y)∂ 2 (x).<br />
2<br />
|a + b| 2 = |a| 2 + |b| 2 + ab + ab.<br />
x=y ∞<br />
If<br />
the last term is equal <strong>to</strong><br />
and the <strong>to</strong>tal in this case is<br />
∂ 1 = ∂ 2 = µ,<br />
(q 2 + q)(q − 2)(q − 1),<br />
so<br />
Clearly,<br />
since<br />
(q + 1)(q − 1)[(q + 1) + (q − 1) + 2q(q − 2)] = (q + 1)(q − 1)2q(q − 1) = 2|G|,<br />
◦ν V1 , 2<br />
, ν V1 , 2<br />
= 2.<br />
C µ ∧ Ind G<br />
B C µ,µ ,<br />
Hom G (C µ , Ind G BC µ,µ ) = Hom B (C µ , C µ ) = C (Theorem 4.33).<br />
There<strong>for</strong>e, Ind G<br />
B C µ,µ = C µ W µ ; W µ is irreducible; and the character of W µ is different <strong>for</strong> distinct<br />
values of µ, proving that W µ are distinct.<br />
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