By Borovik A. V.
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Additional info for 3-local characterization of Held groups
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. Usually it is further required that a spherical function assume the value 1 at 1G . If we wish to emphasize this normalization we will say that ϕ is a normalized spherical function.
Let K be a fixed compact open subgroup and δ an irreducible (smooth) representation of K. We know that δ is finite-dimensional; let v1 , . . , 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.
Spherical representations: elementary results These are the most important representations for applications. However, the present general discussion is insufficient for finer study of spherical representations of totally disconnected groups G. Indeed, we can do nothing further with spherical representations until later when we assume that G is p-adic reductive. Suppose for this section that G is unimodular, and fix a compact open subgroup K in G. 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.