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STABILITY OF CERTAIN OSCILLATORY INTEGRALS 3in particular, we can embed N as a subgroup of ˜G via n ↦→ (n, 1). Consider the spaceΩ gen (N\ ˜G, ψ N ) of smooth genuine functions W on ˜G such that W ((n, 1)g) = ψ N (n)W (g)for all n ∈ N, g ∈ ˜G. Theorems 1 and 2 are also valid for Ω gen (N\ ˜G, ψ N ). In a forthcomingpaper we will use these results to analyze the Whittaker coefficients of cusp forms on themetaplectic group [LM].Acknowledgement. We thank the referee for useful suggestions.2. Notation and preliminariesWe first fix some more notation. Throughout F will be a p-adic field with ring of integersO, normalized absolute value ∣ ∣·∣∣and valuation v. Let G be a semisimple split group overF of rank r. (Theorems 1 and 2 easily reduce to the semisimple case.) Let B be a Borelsubgroup of G, A a maximal F -split torus contained in B and N the unipotent radical ofB so that B = AN. Let B = AN be the opposite Borel subgroup with respect to A. LetK be a hyperspecial maximal compact subgroup of G in good position with respect to B.We have Iwasawa decomposition G = ANK. Let X ∗ (A) (resp. X ∗ (A)) be the lattice ofrational characters (resp. co-characters) of A.2.1. Roots and weights. Let Φ ⊆ X ∗ (A) be the set of roots of A in Lie(G), Φ + thesubset of positive roots and ∆ 0 the subset of simple roots with respect to B. Similarly,let ∆ ∨ 0 ⊆ Φ ∨ + ⊆ Φ ∨ ⊆ X ∗ (A) be the sets of (simple, or positive) co-roots. We denote byα ↔ α ∨ the canonical bijection between Φ and Φ ∨ (resp., Φ + ↔ Φ ∨ +, ∆ 0 ↔ ∆ ∨ 0 ).For any α ∈ Φ let N α be the one-parameter subgroup corresponding to α and choose aparametrization n α : G a → N α defined over F such that n α (O) = N α ∩ K. For any α ∈ ∆ 0we write N α for the unipotent radical of the (semisimple) rank one parabolic subgroupB ∪ Bs α B. Thus, N = N α ⋉ N α . We also have(3) N ∩ K = (N α ∩ K) ⋉ (N α ∩ K).Let a ∗ 0 = X ∗ (A) ⊗ R and a 0 = X ∗ (A) ⊗ R. These are r-dimensional R-vector spaces and∆ 0 (resp., ∆ ∨ 0 ) is a basis for a ∗ 0 (resp., a 0 ). The canonical pairing 〈·, ·〉 : X ∗ (A)×X ∗ (A) → Zextends to a paring a ∗ 0 × a 0 → R. Any λ = ∑ ∣ α∈∆ 0r α α ∈ a ∗ 0 defines a positive character∣ λ : A → R+ given by∣ ∏ ∣ ∣∣ λ∣(t) = ∣α(t) r α.α∈∆ 0By Iwasawa decomposition we extend ∣ ∣ λ to a left-N and right-K invariant function on G.For any compact subset C ⊆ G there exists a constant κ C > 0 (depending also on λ) suchthat∣ ∣ ∣ ∣ ∣(4) κ −1 ∣ λ∣(g) ≤ ∣λ∣(gh) ≤ κC∣λ∣(g)Cfor all g ∈ G, h ∈ C.Let H : A → a 0 be the homomorphism defined by q −〈χ,H(t)〉 = ∣ ∣ χ(t)∣ ∣ for all t ∈ A,χ ∈ X ∗ (A). Thus, H(χ ∨ (r)) = v(r)χ ∨ for any χ ∨ ∈ X ∗ (A).