Inverse Satake isomorphism and change of weight

Let G G be any connected reductive alttext="p"> p encoding="application/x-tex">p -adic group. K subset-of upper K ⊂ encoding="application/x-tex">K\subset G special parahoric subgroup and V comma prime"> V , ′ encoding="application/x-tex">V,V’ two irreducible smooth alttext="ModifyingAbove double-struck F Subscript p Baseline With bar left-bracket right-bracket"> F ¯<!-- ¯ </mml:mover> stretchy="false">[ stretchy="false">] encoding="application/x-tex">\overline {\mathbb {F}_p}[K] -modules. The main goal of this article is to compute the image Hecke bimodule E n d ModifyingAbove Sub right-bracket left-parenthesis c minus I Superscript G prime right-parenthesis"> End ⁡<!-- ⁡ stretchy="false">( c −<!-- − <mml:mi>I n d stretchy="false">) encoding="application/x-tex">\operatorname {End}_{\overline {F}_p}[K]}(c-Ind_K^G V, c-Ind_K^G V’) by generalized Satake transform give an explicit formula for its inverse, using pro-. This immediately implies “change weight theorem” in proof classification mod admissible representations terms supersingular ones. A simpler change theorem, not or Lusztig-Kato formula, given when split appendix quasi-split, almost all K"> encoding="application/x-tex">K