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Notice that for w ∈ W and x ∈ sp, τsp (w)(x) = xw, w = wt Jxw = β(w)(Jx). 8), χx (w) dw = chc(x) = W e2πiw t1 4 Jxw dw = µ ˆ W 1 Jx 4 (x ∈ sp). 9). 10. The case when H ⊆ G is compact In this section the Cartan subgroup H ⊆ G is compact. 1) Vj j∈J be a decomposition of V into H -irreducible subspaces over D. 1) contains the trivial component, which shall be denoted by V0 . There is no (non-zero) trivial component in any other case. 2) x |V j = ix j (x ∈ h , j ∈ J \ {0}). 336 T. 3. 12). Assume p ≥ 0.

Wallach, Real Reductive Groups, I. Academic Press, INC, 1988 [W2] N. Wallach, Invariant differential operators on a reductive Lie algebra and Weyl group representations. J. Amer. Math. Soc. 6 (1993) 779–816 [Wy] H. Weyl, The Classical Groups, their invariants and representations.

Let Γ ⊆ h be an open convex cone. Fix a norm | | on h. For γ > 0 set Z γ = {h · exp(i y); h ∈ X, y ∈ Γ, |y| < γ }. Let f be a holomorphic function on Zγ \ X. Assume that the function f satisfies the following growth condition | f(h · exp(i y))| ≤ const |y|−N , (h ∈ X, y ∈ Γ, |y| < γ). Suppose C is a (n + 1) chain in Zγ , with the boundary ∂C = C0 − C 1 , where C 0 = X and C 1 ⊆ Z γ \ X. Let dz be an invariant holomorphic n-form 360 T. Przebinda on HC . 7) Ψ N (z) f(z) dz − C1 X d (Ψ N (z) f(z) dz) , C where, for N large enough, the integrals on the right hand side are absolutely convergent.

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A Cauchy Harish-Chandra integral, for a real reductive dual pair by Przebinda T.


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