Lie groups in prolongation theory by Van Eck H.N.

By Van Eck H.N.

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By Van Eck H.N.

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Thus, the space π Since an irreducible δ of K is necessarily finite-dimensional, admissibility of a representation π is equivalent to the assertion that the δ-isotypic subspace in π is finite-dimensional, for all irreducibles δ of K. Proofs of the following are left to the reader, as they are but further exercises in application of the smooth representation theory of compact (totally disconnected) groups. Subrepresentations and quotient representations of admissible representations are admissible.

Therefore, for any w ∈ π K there is η ∈ H(G) so that ηv = w Then w = eK w = eK ∗ ηv = eK ∗ η ∗ eK v = Λ(eK ∗ η ∗ eK )v ∈ k · v ♣ This is the desired result. Let , :π×π ˇ −→ k be the canonical k-bilinear pairing, and let cvλ (g) = c(v, λ)(g) = π(g)v, λ be the coefficient function, as usual. Lemma: Let π be a smooth representation of G with a K-spherical vector v = 0. Let λ ∈ (ˇ π ) K be a smooth functional such that λv = 0. Then the k-valued function f on G defined by ϕ(g) = cvλ (g) = π(g)v, λ is a K-spherical function.

The K-spherical Hecke algebra is H(G, K). We consider k-algebra homomorphisms Λ : H(G, K) −→ k where H(G, K) has the convolution algebra structure. D. Groups (July 8, 2005) for all η ∈ H(G, K). A K-spherical vector in a smooth representation π of G is a vector 0 = v ∈ π K which is an eigenvector for H(G, K) with some eigenvalue Λ. A K-spherical function is a k-valued function on G which is left and right K-invariant and which is an eigenvector for H(G, K) under the right-translation action of G on k-valued functions on G.

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