22
GEORGE A. HAGEDORN
to the range of P(X) is represented by a self-adjoint traceless 4 x 4 matrix valued function
M\(X) whose entries are Ck functions that all vanish when X = 0. Because h(X) commutes
with the two projections onto the carrier subspaces of the C and D representations and with
the action of /C, M\{X) commutes with
1 0 0 0'
0 1 0 0
0 0 0 0
0 0 0 0,
and
where D(K?) is multiplication by eu
(Conjugation),
It follows that M\(X) must have the form
° \
/ p{X)
1
{X) + id{X) 0 0
j(X)-i6{X) -P{X) 0 0
0 0 p(X) j(X)-i6(X)
\ 0 0 y(X) + i6(X) -/3(X)
where /?, 7, and 6 are C real valued functions. The difference between E^(X) and Eg(X)
is the same as the difference between the eigenvalues of M\{X). By direct computation, the
eigenvalues of M\(X) are
±
V
//?(X)2 +
7
(X)2 + 8(X)\
Thus, the eigenvalues Ej\[X) and Eg(X) cross precisely at those points X where P(X) =
j(X) S(X) = 0. Generically this defines a codimension 3 submanifold I\ Further-
more, it is clear that the eigenvalues E^(X) and Eg(X) are continuous, but generically not
differentiable near T.
By standard Taylor series results, MX(X) has the form M\{X) = Ni(X) 4- 0(||X|| 2 ),
where
\
I
n.
Y
7/.
Y
—h.Y
n n
Ni(X)
I b-X c-X + id-X 0 0
c-X-id-X -b-X 0 0
0 0 b-X c-X-id-X
0 0 c-X + id-X -b-X
for some vectors 6, c, and d. Generically 6, c, and d are linearly independent. By a rotation
of the coordinate system we may assume that only the first three components of 6, c, and d
are non-zero.
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