C crossings, h(X) commutes with the orthogonal projections Pi and P2 onto the mutually
orthogonal carrier subspaces Hi and H2 associated with U\ and L^, respectively.
For X in a neighborhood of the origin, one can write the spectral projection P{X) for
h(X) associated with the eigenvalues Ej[(X) and Eg(X) as an integral of the resolvent of
h(X). From this it follows that P(X) is a Ck, rank 3 operator valued function of X near
X = 0 that commutes with Pi and Pi- Since U\ and U2 are inequivalent, it follows that
PA(X) = PiP(X) and Pjs(X) = P2P{X) are Ck, rank one and (respectively) rank two
orthogonal projections that project onto mutually orthogonal subspaces.
We construct a Ck unit-vector valued function $A(') exactly as in the case of a Type
C crossing, so that K$j±{X) ^j^(X). For a Type F crossing we choose 3g i(0) to be an
arbitrary unit vector in the range of P#(0). We then let
* ^ P
*B,2(X) = K*B,lW.
Because K is antiunitary and D{K?) 1, it follows that $&i(X) and ^g^G^O comprise
an orthonormal basis for the range of Pg(X).
For Type G crossings, we let PQ and P p denote the orthogonal projections onto the
carrier subspaces for the two representations C and D of the subgroup H that are involved.
These projections commute with P&{X) and project onto mutually orthogonal subspaces.
Furthermore, P(jPg(X) and Pj^P^{X) are rank one projections. We choose £# i(0) to be a
unit vector in the range of P D P # ( 0 ) . We then let
* ^ PB(X)*B,I(P)
yJ(*B,l(0), PB(X)*Btl(0))
*B,2(X) = £ * 3 , l W -
From the structure theory of type III corepresentations, it follows that $B,2(X) belongs to
the range of PQPB(X), and that $gi(X) and $ # 2 p 0 comprise a Ck orthonormal basis
for the range of Pg(X).
Let h^-(X) denote the restriction of h(X) to the subspace orthogonal to the range
of P(X). By using ^ ( X ) , $ g i(X) and $ # 2 p 0 as a basis for the range of P(X) and
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