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By Przebinda T.

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Part (a) of the proposition is immediate from the following lemma. 4. The set SMm+(C) is contractible. 334 T. Przebinda Proof (Rossmann). For a > 0, consider the map: A = B + iC → A˜ = B + a C 2 + iC ∈ SMm (C). SMm (C) Then A ∈ SMm+(C) if and only if there is a > 0 such that the real part of A˜ is positive definite. Since the set of complex symmetric matrices with a positive definite real part is convex, we are done. 4) implies that there is a unique holomorphic function SMm+(C) A → det 1/2 (A) ∈ C, which coincides with the positive square root of the determinant of A, if A is real and positive definite.

We give orientations to CS , C S (1), and C SL |α , by declaring the following charts to be positive: C SL 1+n x + itySL → (t, κ ◦ c−1 , S (x)) ∈ R C SL (1) C SL |α n x + itySL → κ ◦ c−1 S (x) ∈ R , n x + itySL → (t, 0) + κ ◦ c−1 S (x) ∈ R . n Let hiSr = {x ∈ hS , α ◦ c−1 S (x) = 0 for all α ∈ ΦS }. By a theorem of Harish-Chandra, [Va, part I, p. 16), extends to a smooth function on hiSr , which shall be denoted by the same 340 T. Przebinda symbol ψS . 1)). 19. Fix an element x ∈ h r . Then for any integer N ≥ 0, large enough, and for all ψ ∈ S(g), P(x )πh (x )  = nS  g chc(x + x)ψ(x) dx C LS (1)  ν L,x ,S,N − C LS d(ν L,x ,S,N ) − α∈ΦnS C LS |α ν L,x ,S,N  , where νL,x ,S,N = FL,x ◦ c−1 S · ψ S,N · µ S , the unmarked summation is over a maximal family of mutually non-conjugate Cartan subalgebras hS ⊆ g and over all injections L : J \ {0} → J \ {0}.

9) 0 = w0 (x )k w∗0 + x1 (x )k y1∗ + x2 (x )k y2∗ + ... ) . Since x ∈ h r is regular, the odd powers (x )k span h over the field of the points in the center of D, fixed by the involution. 9) holds with the (x )k replaced by an arbitrary element of h . 10) w0 w∗0 = 0, x j y∗j = 0, ( j ≥ 1). 10) means that the image of w∗0 is an isotropic subspace of Vc . 8), this image is preserved by x . Hence, by the classification of Cartan subalgebras in g , w0 = 0. 8), for pairs of type II, we check that sj = 0 for j ≥ 1.

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