By Przebinda T.

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32) k be the decomposition of V into R[˜xs ]-isotypic subspaces over D. 32) is direct, orthogonal, and the sets of eigenvalues of˜xs |V˜k are ˜ j,k = Hom(V j , V˜k ). Since the x ∈ h r is regular, disjoint, as k varies. , m. Ker(x + x˜s ) ∩ W A straightforward, case by case, verification shows that for 1 ≤ k ≤ m, ˜ ˜ k,k = Ker(x + x˜s ) ∩ Wk,k W ˜ k,k ⊕ (Ker(x + x˜s ) ∩ W ˜ k,k )⊥ Ker(x + x˜s ) ∩ W if D = C, if D = C. 34) W = Ker(x + x˜s ) ⊕ (Ker(x + x˜s ))⊥ . Using the complex structure i on Vk we view Ker(x + x˜s )∩ W˜ k,k as a complex vector space.

1). 14) proves the following proposition. 3. For any Ψ ∈ Cc∞ (G) G˜ Chc(h g)Ψ(g) dg −1/2 = δP (h ) . G L(X )/Hs ˜ ) G(U ChcWc (h c h)Ψ P (gh s g−1 h) dh d g, where Hs = H | X , is the restriction of H to X , h c = h |Vc , h s = h | X , and Wc = Hom(Vc , U ). From now on we assume that the pair G, G is of type I, and that the Cartan subgroup H ⊆ G is compact. 356 T. Przebinda Let H ⊆ G be a compact Cartan subgroup. a), uniquely extends to a rational function on H˜ H˜ C . 4) ChcW (h h) = (h ∈ H˜ C , h ∈ H˜ C ), ChcW j,k (h h) j∈J , k∈J where the subscript (W or Wj,k ) indicates the symplectic space with respect to which the corresponding function is defined.

356 T. Przebinda Let H ⊆ G be a compact Cartan subgroup. a), uniquely extends to a rational function on H˜ H˜ C . 4) ChcW (h h) = (h ∈ H˜ C , h ∈ H˜ C ), ChcW j,k (h h) j∈J , k∈J where the subscript (W or Wj,k ) indicates the symplectic space with respect to which the corresponding function is defined. 5) F˜L,h (h) = P˜L (h )Chc L (h h), ChcL (h h) = ChcW j,L( j) (h h), j∈J \{0} where the summation is over all injections L : J \ {0} → J \ {0}, each P˜L is a regular function on H˜ C , h ∈ H˜ C , and h ∈ H˜ C .

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


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