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Example text

3) W j = Hom(X j , V ) ⊕ Hom(Y j , V ) ( j ≥ 1). 4) a = sp(Wc ) ⊕ EndR (Hom(X 1 , V )) ⊕ EndR (Hom(X 2 , V )) ⊕ ... A = Sp(Wc ) × G L R (Hom(X 1 , V )) × G L R (Hom(X 2 , V )) × ... Let A be the centralizer of A in Sp. Let A j be the restriction of A to X j . Then A j is isomorphic to G L1 (R). The restriction of A to Wc is isomorphic to O1 , the two element group. 5) (Sp2n (R), O1 )×(G L n1 (R), G L 1 (R))×(G L n2 (R), G L 1 (R))×... Let W A = (Wc \ {0}) × W1 A1 × W2 A2 × .... 346 T. Przebinda .

G /H G˜L(U ) Ψ P (gh g−1 h) dh d g. Let G, G be a pair of type I. 13). Let P ⊆ G˜ be the parabolic subgroup preserving X , and let P ⊆ G˜ be the parabolic subgroup preserving X . Let N ⊆ P and N ⊆ P be the unipotent radicals. The Levi factor of P, M, coincides with the double cover of G L(X ) · G(U ), where G(U ) is the restriction of G to U. 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 ).

352 T. Przebinda The group G L C (W ) is a complexification of U. Let G L C (W ) = {(g, h); ++ g ∈ G L C (W ), h 2 = det(g)}, and let G L C (W ) = {(g, h); g ∈ G L C (W ), g∗ g < 1, h 2 = det(g)}. A straightforward calculation shows that the map G L ++ C (W ) (g, h) → g, h det(1 − g) ∈ U˜ C preserves multiplication. 13), extends to a rational function on G L C (W ). 13. The pair G L n (R), G L 1 (R) Here we use the notation developed in Sect. 4. 1. For any Ψ ∈ Cc∞ (G), T(Ψ) is a function on Wmax such that .