Hermitian Analogue of a Theorem of Springer - Chennai ...

HERMITIAN ANALOGUE OF A THEOREM OF SPRINGER 783
3. HERMITIAN FORMS OVER QUATERNION ALGEBRAS
We begin by recalling Ž�S �, cf. �PSS�. an exact sequence of Witt groups of
hermitian forms over quaternion algebras.
Let H be a quaternion algebra over k and let � be the canonical
involution on H. Let L � kŽ �. be a maximal commutative subfield of H,
with � 2 � a � k*. Let � � H be such that �� ���� and � 2 � b � k*.
Then H � L � �L. Let � be the non-trivial automorphism over L over
0
k. We have the L-linear projections
� 1: H � L, �1Ž ���� . � � ,
� 2: H � L, �2Ž ���� . � �
for �, � � L. Ifh: V � V � H is an �-hermitian form over Ž H, � . ,we
define h : V � V � L, by h Ž x, y. � � ŽhŽ x, y ..
i i i . Then h � h1��h 2.
Since hŽ x�, y. � �Ž �. hŽ x, y. ���hŽ x, y . , we have h Ž x, y. 2 �
�1 �b h Ž x�, y . . It is easy to see that � Ž h. 1 1 � h 1:
V � V � L is an
�-hermitian form over L and that � Ž h. 2 � h 2:
V � V � L is an ��-symmetric
form over L. Further, � and � induce homomorphisms Žcf. �
1 2
S,
PSS�.
Let
� : W � H, � � W � 1 Ž . Ž L, � 0.
� : W � H, � � W�� 2 Ž . Ž L . .
� : W L, � � W�1 Ž 0 . Ž H, � .
be the homomorphism defined as follows: Let f: V � V � L be a hermitian
space over Ž L, � . . Write V � H � V � V�. Define �Ž f .
0 L : V �L H �
V �L H � H by
� Ž f.Ž x1�y1�, x2�y2 �.
Ž .
� � fŽ x , x . � fŽ x , y . � � � fŽ y , x . � � fŽ y , y . � ,
1 2 1 2 1 2 1 2
for x , x , y , y � V. Then we have the following
1 2 1 2
Ž � �.
THEOREM 3.1 cf. S, PSS . With notation as abo�e, the sequence
� 1 � � 2
�1
0
0 � WŽ H, � . � WŽ L, � . � W Ž H, � . � WŽ L.
is exact.
With the notation as above, we have the following lemma.