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.