fix: Satake transformation -> Satake transform

Author: vEnhance

thesis/satake.tex

@@ -8,8 +8,8 @@ \section{Base change}
 with order $n!$
 (since $S_n(F) \subseteq \GL_n(E)$ is already reserved for the symmetric space).
 
-\subsection{Background on the Satake transformation in general}
-We recall a general form of the Satake transformation, which will be used later.
+\subsection{Background on the Satake transform in general}
+We recall a general form of the Satake transform, which will be used later.
 
 For this subsection, $G$ will denote an arbitrary connected reductive group
 over some non-Archimedean local field $F$.
@@ -50,7 +50,7 @@ \subsection{Background on the Satake transformation in general}
 and once when $G$ is a unitary group.
 
 
-\subsection{The Satake transformation for the particular Hecke algebras $\HH(\GL_n(E))$ and $\HH(\U(\VV_n^+))$}
+\subsection{The Satake transform for the particular Hecke algebras $\HH(\GL_n(E))$ and $\HH(\U(\VV_n^+))$}
 To take the Satake transform of $\HH(\U(\VV_n^+))$, we define the following abbreviations.
 \begin{itemize}
   \ii Let $T$ denote the split diagonal torus of $\GL_n$.
@@ -95,11 +95,11 @@ \subsection{The Satake transformation for the particular Hecke algebras $\HH(\GL
     %            & $\left< \frac{m-1}{2}, \frac{m-3}{2}, \dots, -\frac{m-1}{2} \right>$??
     \bottomrule
   \end{tabular}
-  \caption{Data needed to run the Satake transformation.}
+  \caption{Data needed to run the Satake transform.}
   \label{tab:satakestuff}
 \end{table}
 
-Hence, the Satake transformations obtained can be viewed as
+Hence, the Satake transforms obtained can be viewed as
 \begin{align*}
   \Sat &\colon \HH(\GL_n(E)) \xrightarrow{\sim} \QQ[T(E) / T(\OO_E)]^{\Sym(n)} \\
   \Sat &\colon \HH(\U(\VV_n^+))\xrightarrow{\sim} \QQ[A(F) / A(\OO_F)]^{W_m}
@@ -132,11 +132,11 @@ \subsection{The Satake transformation for the particular Hecke algebras $\HH(\GL
   \Sat &\colon \HH(\U(\VV_n^+)) \xrightarrow{\sim} \QQ[Y_1^{\pm}, \dots, Y_m^{\pm}]^{W_m}.
 \end{align*}
 
-\subsection{Relation of Satake transformation to base change}
+\subsection{Relation of Satake transform to base change}
 Let
 \[ \BC \colon \HH(\GL_n(E)) \to \HH(\U(\VV_n^+)) \]
 denote the stable base change morphism from $\GL_n(E)$ to the unitary group $\U$.
-The relevance of the Satake transformation is that
+The relevance of the Satake transform is that
 (see e.g.\ \cite[Proposition 3.4]{ref:leslie})
 it gives a way to make this $\BC$ completely explicit:
 we have a commutative diagram