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By Hanspeter Kraft

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V2m ) of V with corresponding matrix J , see (∗). 3 Exe. ex CHAPTER II. ALGEBRAIC GROUPS w , v3 , . . , v2m ) of V with corresponding matrix J . Hence the map g : vi → vi belongs to Sp(V ), and the claim follows. Since the function β−1 is irreducible, the subset P ⊆ V ×V is an irreducible subvariety. 4 below applied to G := Sp(V ) ⊆ GL(V ⊕ V )). This implies that for every g ∈ Sp(V ), there is an h ∈ Sp(V )◦ such that hg(v) = v and hg(w) = w. Therefore, hg belongs to the intersection if the two stabilizers of v and w.

Such a basis (v1 , v2 , . . , we have q(vi , vj ) = δij . It follows that O(V, q) is isomorphic to the classical orthogonal group On := On (C) := {g ∈ GLn | g t g = En }, special orthogo- and that any two orthogonal groups O(V, q), O(V, q ) are conjugate in GL(V ). Furthermore, the special orthogonal group is defined in the following way: nal group SOn := SOn (C) := On ∩ SLn , SO(V ) := SO(V, q) := O(V, q) ∩ SL(V ). −1 SOn , SO(V ) SO(V, q), We have On = SOn ∪ 1 .. SOn , and so On / SOn Z/2Z. 1.

We have q(vi , vj ) = δij . It follows that O(V, q) is isomorphic to the classical orthogonal group On := On (C) := {g ∈ GLn | g t g = En }, special orthogo- and that any two orthogonal groups O(V, q), O(V, q ) are conjugate in GL(V ). Furthermore, the special orthogonal group is defined in the following way: nal group SOn := SOn (C) := On ∩ SLn , SO(V ) := SO(V, q) := O(V, q) ∩ SL(V ). −1 SOn , SO(V ) SO(V, q), We have On = SOn ∪ 1 .. SOn , and so On / SOn Z/2Z. 1. Exercise. Describe O(V, q) and SO(V, q) for V := C2 and q(x, y) := xy.

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Algebraic Transformation Groups: An Introduction by Hanspeter Kraft


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