fix: typos from Mark Sellke (thanks!)

Author: vEnhance

Diff for: thesis/acknowledgments.tex

@@ -24,6 +24,9 @@ \chapter*{Acknowledgments}
 for inviting me to speak about my work-in-progress, and their helpful comments on it
 leading directly to improvements to this thesis.
 
+I thank Mark Sellke for proofreading a draft of this thesis
+and finding several corrections.
+
 Finally, I thank Zhiwei Yun and Ben Howard for serving on my thesis committee,
 as well as David Vogan and Nike Sun for serving on my qualifying exams committee.
 

Diff for: thesis/bg.tex

@@ -186,7 +186,7 @@ \section{The spaces $S_n(F)$ and $S_n(F) \times V'_n(F)$}
   \[ V'_n(F) \coloneqq F^n \times (F^n)^\vee \]
   where $-^\vee$ denotes the $F$-dual space, i.e., $(F^n)^\vee = \Hom_F(F^n, F)$.
   Then we may also consider the augmented space
-  \[ S_n(F) \times V'_n(F) \]
+  \[ S_n(F) \times V'_n(F). \]
   If we identify $F^n$ with column vectors of length $n-1$ and $(F^n)^\vee$
   with row vectors of length $n$ then we have a left action of $\GL_n(F)$ by
   \begin{align*}
@@ -250,7 +250,7 @@ \section{The specific Hecke algebras $\HH(\GL_n(E))$ and $\HH(\U(\VV_n^+))$ and
 that will come up consistently:
 \begin{align*}
   \HH(\GL_n(E)) &\coloneqq \HH(\GL_n(E), \GL_n(\OO_E)) \\
-  \HH(\U(\VV_n^+)) &\coloneqq \HH(\U(\VV_n^+), \U(\VV_n^+) \cap \GL_n(\OO_E))
+  \HH(\U(\VV_n^+)) &\coloneqq \HH(\U(\VV_n^+), \U(\VV_n^+) \cap \GL_n(\OO_E)).
 \end{align*}
 Note that $\GL_n(\OO_E)$ and $\U(\VV_n^+) \cap \GL_n(\OO_E)$
 are the natural hyperspecial maximal compact subgroups of $\GL_n(E)$ and $\U(\VV_n^+$),

Diff for: thesis/match.tex

@@ -147,10 +147,11 @@ \section{Matching in the group version of the inhomogeneous AFL}
 We write the following abbreviation:
 \begin{definition}
   [$S_n(F)\rs^\pm$]
-  We let \[ S_n(F)\rs^- \]
-  denote the set of subset of those orbits $[S_n(F) \times V'_n(F)]\rs$
-  which with an element of $[\U(\VV_n^-) \times \VV_n^-]\rs$.
+  We let \[ S_n(F)\rs^- \subset S_n(F)\rs \]
+  denote the subset of elements in $S_n(F)\rs$ that match
+  with an element in $\U(\VV_n^-)\rs$.
   Define $S_n(F)\rs^+$ similarly.
+  Hence $S_n(F)\rs = S_n(F)\rs^- \amalg S_n(F)\rs^+$.
 \end{definition}
 
 \section{Matching in the semi-Lie version of the AFL}
@@ -194,9 +195,9 @@ \section{Matching in the semi-Lie version of the AFL}
 Accordingly we write the following shorthand:
 \begin{definition}
   [$(S_n(F) \times V'_n(F))\rs^\pm$]
-  We let
-  \[ (S_n(F) \times V'_n(F))\rs^- \]
-  denote the set of subset of those orbits $[S_n(F) \times V'_n(F)]\rs$
-  which with an element of $[\U(\VV_n^-) \times \VV_n^-]\rs$.
+  We let \[ (S_n(F) \times V'_n(F))\rs^- \subset (S_n(F) \times V'_n(F))\rs  \]
+  denote the subset of those elements in $(S_n(F) \times V'_n(F))\rs$ that match
+  with an element of $(\U(\VV_n^-) \times \VV_n^-)\rs$.
   Define $(S_n(F) \times V'_n(F))\rs^+$ analogously.
+  Hence $(S_n(F) \times V'_n(F))\rs = (S_n(F) \times V'_n(F))\rs^- \amalg (S_n(F) \times V'_n(F))\rs^+$.
 \end{definition}

Diff for: thesis/orb0.tex

@@ -51,7 +51,7 @@ \section{Basis for the indicator functions in $\HH(S_3(F))$}
 \label{ch:orbital0_hecke_basis}
 From now on assume $n = 3$.
 We have the symmetric space
-\[ S_3(F) \coloneqq \left\{ g \in \GL_3(E) \mid g \bar{g} = \id_3 \right\}. \]
+\[ S_3(F) \coloneqq \left\{ g \in \GL_3(E) \mid g \bar{g} = \id_3 \right\} \]
 which has a left action under $\GL_3(E)$ by $g \cdot s \mapsto gs\bar{g}\inv$.
 
 Then $S_3(F)$ admits the following decomposition, which we will use:
@@ -243,7 +243,7 @@ \section{Parameters used in the calculation of the weighted orbital integral}
   \ell \coloneqq v(b^2 - 4 a \ol d).
   \label{eq:ell}
 \end{equation}
-We will also define one additional parameter useful when $\ell$ is even
+We will also define one additional parameter which is useful when $\ell$ is even
 (but as we will see, redundant for odd $\ell$):
 \begin{equation}
   \lambda \coloneqq v(1-u \bar u) \equiv 1 \pmod 2.
@@ -590,7 +590,7 @@ \subsection{Full explicit weighted orbital integral for the case where $\ell$ od
   \[
     \Orb(\gamma, \mathbf{1}_{K'_{S, \le r}}, s)
     = \sum_{k = -2r}^{\ell + 2\delta + 2r}
-    (-1)^k \left( 1 + q + q^2 + \dots + q^{\nn_{\gamma}(k)}  \right) (q^s)^k
+    (-1)^k \left( 1 + q + q^2 + \dots + q^{\nn_{\gamma}(k)} \right) (q^s)^k.
   \]
 \end{restatable}
 \begin{remark}
@@ -815,7 +815,7 @@ \subsection{Full explicit weighted orbital integral for the case $\ell < 0$}
     \Orb(\gamma, \mathbf{1}_{K'_{S, \le r}}, s)
     &= \sum_{k = -2r}^{\lambda + 2r-4|v(d)|}
     (-1)^k \left( 1 + q + q^2 + \dots + q^{\nn_{\gamma}(k)} \right) (q^s)^k \\
-    &+ \sum_{k = -r-|v(d)|}^{\lambda+r-3|v(d)|} \cc_\gamma(k) (-1)^k q^{r-|v(d)|} (q^s)^k \\
+    &+ \sum_{k = -r-|v(d)|}^{\lambda+r-3|v(d)|} \cc_\gamma(k) (-1)^k q^{r-|v(d)|} (q^s)^k.
   \end{align*}
 \end{restatable}
 

Diff for: thesis/orb2.tex

@@ -434,7 +434,7 @@ \section{Merging of $I_{n \le 0}$ with $I_{n > 0}^{\text{1+2}}$ (and proof of \C
       \left\lfloor \frac{k+r}{2} \right\rfloor + r & \text{if }{-r} \le k \le \ell-r \\
       \frac{\ell}{2} + r - (k-r) \% 2 & \text{if } \ell - r \le k \le 2\delta + r \\
       \left\lfloor \frac{(\ell+2\delta+r)-k}{2} \right\rfloor + r & \text{if } 2\delta + r \le k \le 2\lambda - \ell + 2\delta + r \\
-      (\lambda + 2\delta + 2r) - k & \text{if } 2 \delta + 2 \lambda - \ell + r \le k \le \lambda + 2\delta + 2r.
+      (\lambda + 2\delta + 2r) - k & \text{if } 2 \delta + 2 \lambda - \ell + r \le k \le \lambda + 2\delta + 2r
     \end{cases}
   \]
   in the case $\lambda \le \ell+r$.
@@ -1602,7 +1602,7 @@ \subsection{Analysis of the $\cc_\gamma$ sum}
     \begin{aligned}
       &\sum_{k=-r}^{W-r-1} (-1)^{k} \cdot k \cdot (k+r)
       + \sum_{k=W+L+r+1}^{2W+L+r} (-1)^{k} \cdot k \cdot (2W+L+r-k) \\
-      &= (-1)^{W+r} \frac{W(L+2r+2)}{2} - (-1)^{W+r} (W\%2) \cdot \frac{2W+L+2r}{2}
+      &= (-1)^{W+r} \frac{W(L+2r+2)}{2} - (-1)^{W+r} (W\%2) \cdot \frac{2W+L+2r}{2}.
     \end{aligned}
     \label{eq:moment15}
   \end{equation}

Diff for: thesis/orbFJ1.tex

@@ -121,7 +121,7 @@ \subsection{The quadratic constraint}
 As before, we complete the square:
 \[
   -y^2+ \frac{d-a}{c} y + \frac bc
-  = -\left( y - \frac{d-a}{2c} \right)^2 + \frac bc + \frac{(d-a)^2}{4c^2}
+  = -\left( y - \frac{d-a}{2c} \right)^2 + \frac bc + \frac{(d-a)^2}{4c^2}.
 \]
 Because $b \bar c = 1 - a \bar a$ has odd valuation,
 it follows that $\frac b c = \frac{1-a \bar a}{c\bar c}$ has odd valuation to.

Diff for: thesis/orbFJ2.tex

@@ -45,7 +45,7 @@ \section{The contribution for Case 5}
 \end{align*}
 We'll change the summation variable to
 \[ k \coloneqq 2n_2 - m + v(c) + r
-\iff m = 2n_2 - k + v(c) + r \]
+  \iff m = 2n_2 - k + v(c) + r. \]
 Then
 \begin{align*}
   I^{\text{5}}
@@ -239,7 +239,7 @@ \subsection{Sub-case where $v(e) > \frac{\theta}{2} + r$}
   \sum_{k = \theta - v(b) + r + 1}^{2v(e) - \theta + v(c) - r - 1}
   \bigg( 1 + \min\left( v(e), k - \frac{\theta}{2} + v(b) \right) \\
     &\hspace{16ex} - \max\left( \frac{\theta}{2} + r + 1,
-      \left\lceil \frac{k + \theta - v(c) + r + 1}{2} \right\rceil \right) \bigg) (-q^s)^k
+      \left\lceil \frac{k + \theta - v(c) + r + 1}{2} \right\rceil \right) \bigg) (-q^s)^k.
 \end{align*}
 It is natural to split this sum into $k \le v(c) + r$ and $k > v(c) + r$.
 In the former case, we have both $k - \frac{\theta}{2} + v(b) \le v(e)$

Diff for: thesis/satake.tex

@@ -208,7 +208,7 @@ \section{The map $\BC^{\eta^{n-1}}_{S_n}$}
 \section{Calculation of $\BC_{S_n}$ when $n = 3$}
 The goal of this section is to make the base change
 fully known in the special case $n = 3$, where $m = \left\lfloor n/2 \right\rfloor = 1$.
-(In this case $\BC_{S_n}^{\eta^{n-1}} = \BC_{S_n}$ as $\eta^2 = 1$).
+(In this case $\BC_{S_n}^{\eta^{n-1}} = \BC_{S_n}$ as $\eta^2 = 1$.)
 The completed result is \Cref{prop:BC_S3}.
 
 This calculation parallels the $n = 2$ case that was done in \cite[Lemma 7.1.1]{ref:AFLspherical}.
@@ -231,7 +231,7 @@ \subsection{Overview}
 \[ \mathbf{1}_{K'\varpi^{(n_1, n_2, n_3)}K'} \]
 is given by triples of integers $n_1 \ge n_2 \ge n_3 \ge 0$, and is much larger.
 So explicit calculations for the $\rproj_\ast$ or the Satake transforms viewed in
-$\CC[X_1, X_2, X_3]^{\Sym(n)}$ is nontrivial if one works with the entire basis.
+$\CC[X_1, X_2, X_3]^{\Sym(n)}$ are nontrivial if one works with the entire basis.
 
 Hence the overall strategy, to reduce the amount of work we have to do,
 is to focus on only the $\ZZ$-indexed elements