Character expansions and unrefined minimal K-types

Let k be a p-adic field. Let G be the group of k-rational points of a connected reductive group G defined over k, and let [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] be its Lie algebra. Under certain hypotheses on G and k, we get a new character expansion, called a Γ-asymptotic expansion, for representations of positive depth [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]. This character expansion is valid on the G-domain [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /], which is bigger than that where the Harish-Chandra-Howe character expansion is known to be valid.