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By Przebinda T.

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J. Func. Anal. 70 (1987), 1–79 [D-P1] A. Daszkiewicz, T. Przebinda, The Oscillator Correspondence of Orbital Integrals, for pairs of type one in the stable range. Duke Math. J. 82 (1996) 1–20 [D-P2] A. Daszkiewicz, T. Przebinda, The Oscillator Character Formula, for isometry groups of split forms in deep stable range. Invent. math. 123 (1996) 349–376 [D-P3] A. Daszkiewicz, T. Przebinda, A Cauchy Harish-Chandra Integral for the pair u p,q , u1 . In preparation [He] S. Helgason, Groups and Geometric Analysis.

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. Then for any ψ ∈ S(g) and any > 0, the following integral h r (|x j | + 1) p− πh (x ) j∈J \{0} g chc(x + x)ψ(x) dx dx is convergent and defines a continuous seminorm on S(g). 3) we need some preparation. Let H ⊆ G be a compact Cartan subgroup, with the Lie algebra h ⊆ g. 4) W j,k , W j,k ⊆ Hom(V j , V ).

By a theorem of Harish-Chandra, [Va, part II, p. 9), extends to a smooth function on H˜ Si r , which shall be denoted by the same symbol ΨS . 6)). 11. Fix an element h ∈ H˜ r . Then for any integer N ≥ 0, large enough, and for all Ψ ∈ Cc∞ (G˜ 1 ), ∆(h ) = G˜ Chc(h g)Ψ(g) dg  nS  L C˜ S (1)  ν˜ L,x ,S,N − L C˜ S d(˜ν L,x ,S,N ) − α∈ΦnS L C˜ S |α ν˜ L,x ,S,N  , where ν˜ L,x ,S,N = F˜L,x ◦ C −1 ˜ S , the unmarked summation is over S · Ψ S,N · µ a maximal family of mutually non-conjugate Cartan subgroups HS , and over all injections L : J \ {0} → J \ {0}.