By Bhartendu Harishchandra, J.G.M. Mars
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Additional info for Automorphic Forms on Semisimple Lie Groups (Lecture Notes in Mathematics)
The left and right K-invariance follows from the K-invariance of v and λ, by elementary properties of the coefficient functions. D. Groups (July 8, 2005) Note that there is some λ ∈ π ∗ with λ v = 0. Then (π ∗ (eK )λ )(v) = λ (eK v) = λ v = 0 and π ∗ (eK )λ is certainly in π ˇ. ♣ Lemma: Assume that we have given G Haar measure so that meas (K) = 1. Given a k-algebra homomorphism Λ : H(G, K) −→ k, there is at most one K-spherical function ϕ such that ϕ(1) = 1 and Rη ϕ = Λ(η)ϕ for all η ∈ H(G, K).
Groups (July 8, 2005) ♣ by the first computation. Proposition (Godement): Let Λ : H(G, K) −→ k be a k-vectorspace map. Define a k-valued function ϕ on G by ϕ(g) = Λ(eKgK ) Then Λ is a k-algebra map if and only if the functional equation ϕ(g1 θg2 ) dθ = ϕ(g1 ) ϕ(g2 ) × meas (K) K holds. If this does hold, the ϕ is the unique normalized K-spherical function associated to Λ. Proof: Suppose the functional equation holds. The functions eKgK certainly span H(G, K) over k. We have (eKgK ϕ)(h) = ϕ(hx) dx / meas (KgK) = KgK ϕ(hθ1 gθ2 ) dθ1 dθ2 / meas (K)2 = = K K ϕ(hθg) dθ / meas (K) = K since ϕ is right K-invariant.
Vn be a k-basis. D. Groups (July 8, 2005) K = i Kvi is a compact open subgroup inside K. Then K contains another compact open subgroup K which is normal in K: let K act (on the left) on the space K/K of cosets kK , and take K to be the subgroup of K fixing every coset kK . Then K acts trivially on the representation space of δ, so π δ is contained in the set π K of K -fixed vectors. For the converse, suppose that K is a fixed compact open subgroup and that for every irreducible δ of K the δ-isotype in π is of finite multiplicity.