\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{Hecke algebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{generalized_hecke_algebras}{Generalized Hecke algebras}\dotfill \pageref*{generalized_hecke_algebras} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references_and_links}{References and links}\dotfill \pageref*{references_and_links} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\textbf{Hecke algebra} is a term for a class of [[algebras]]. They often appear as [[convolution algebras]] or as double [[coset spaces]]. For p-adic algebraic groups Hecke algebras often play a role similar a [[Lie algebra]] plays in the complex case (the Lie algebra still exists, but is too small).

Typically the term refers to an algebra which is the [[endomorphisms]] of a [[permutation representation]] of a [[topological group]], though some liberties have been taken with this definition, and often the term means some modification of such an algebra.

For example:

\begin{itemize}%
\item If we consider the [[general linear group]] $GL_n(\mathbb{F}_q)$ acting on the set of [[flag|complete flags]] in $\mathbb{F}_q^n$, then we obtain an algebra generated by the endomorphism $\sigma_i$ which sends the characteristic function of one flag $\mathbf{F}=\{F_1\subset \cdots \subset F_{n-1}\subset \mathbb{F}_q^n\}$ to the characteristic function of the set of flags $\mathbf{F}'$ with $F_j'=F_j$ for all $j\neq i$ and $F_i'\neq F_i$. These elements satisfy the relations\begin{displaymath}
\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}
\end{displaymath}
\begin{displaymath}
\sigma_i\sigma_j=\sigma_j\sigma_i \qquad (|i-j|\gt1)
\end{displaymath}
\begin{displaymath}
\sigma_i^2=(q-1)\sigma_i+q
\end{displaymath}

\item If we look at $GL_n(\mathbb{F}_q(\!(t)\!))$ acting on the set of $\mathbb{F}_q[\![t]\!]$ [[lattices]] in $\mathbb{F}_q(\!(t)\!)^n$, then we will obtain the \textbf{spherical Hecke algebra}.
\item A variant of the Hecke algebra is the \textbf{degenerate affine Hecke algebra} of type $A$; this is a deformation of the semidirect product of the symmetric group $S_n$ with the polynomial ring in $n$ variables. The generators are $S_n$ and $y_1, \dots, y_n$, with relations $\sigma y_i \sigma^{-1} = y_{\sigma(i)}$ and $[y_i,y_j] = \sum_{k \neq i,j} (k i j)-(k j i)$; one can replace the $y_i$`s with commuting $x_i$'s with slightly messier relations. As [[George Lusztig]] showed, the representation theory of the affine Hecke algebra is related to the graded or degenerate case.
\item There is a geometric construction of the representations of Weyl algebras when realized as certain Hecke convolution algebras by [[Victor Ginzburg]].

\end{itemize}
\hypertarget{generalized_hecke_algebras}{}\subsection*{{Generalized Hecke algebras}}\label{generalized_hecke_algebras}

To each [[Coxeter group]] $W$ one may associate a Hecke algebra, a certain [[deformation]] of the [[group algebra]] $k[W]$ over a [[field]] $k$, as follows. $W$ is [[group presentation|presented]] by generators $\langle s_i \rangle_{i \in I}$ and relations

\begin{displaymath}
(s_i s_j)^{m_{i j}} = 1
\end{displaymath}
where $m_{i j} = m_{j i}$ and $m_{i i} = 1$ for all $i, j \in I$. The relations may be rewritten:

\begin{displaymath}
s_{i}^{2} = 1, \qquad s_i s_j \ldots = s_j s_i \ldots
\end{displaymath}
where each of the words in the second equation alternate in the letters $s_i$, $s_j$ and has length $m_{i j}$, provided that $m_{i j} \lt \infty$. The corresponding Hecke algebra has basis $W$, and is presented by

\begin{displaymath}
s_{i}^{2} = \frac{q-1}{q} s_i + \frac1{q}, \qquad s_i s_j \ldots = s_j s_i \ldots
\end{displaymath}
These relations may be interpreted structurally as follows (for simplicity, we will consider only finite, aka spherical Coxeter groups). A Coxeter group $W$ may be associated with a suitable [[BN-pair]]; the classical example is where $G$ is an [[algebraic group]], $B$ is a [[Borel subgroup]] (maximal solvable subgroup), and $N$ is the normalizer of a [[maximal torus]] $T$ in $G$. Such $G$ typically arise as automorphism groups of thick $W$-buildings, where $B$ is a stabilizer of a point of the building. The coset space $G/B$ may then be interpreted as a space of flags for a suitable geometry. The Coxeter group itself arises as the quotient $W \cong N/T$, and under the BN-pair axioms there is a well-defined map

\begin{displaymath}
W \to B\backslash G/B: w \mapsto B w B
\end{displaymath}
which is a bijection to the set of double cosets of $B$. (In particular, the double cosets do not depend on the coefficient ring $R$ in which the points $G(R)$ are instantiated.)

When one takes points of the algebraic group $G$ over the coefficient ring $\mathbb{F}_q$, a finite field with $q$ elements, the flag manifold $G_q/B_q \coloneqq G(\mathbb{F}_q)/B(\mathbb{F}_q)$ is also finite. One may calculate

\begin{displaymath}
\itexarray{
\hom_{k[G_q]}(k[G_q/B_q], k[G_q/B_q]) & \cong & k[G_q/B_q]^\ast \otimes_{k[G_q]} k[G_q/B_q] \\
 & \cong & k[B_q\backslash G_q] \otimes_{k[G_q]} k[G_q/B_q] \\
 & \cong & k[B_q\backslash G_q \otimes_{G_q} G_q/B_q] \\
 & \cong & k[B_q \backslash G_q / B_q]
}
\end{displaymath}
so that the double cosets form a linear basis of the algebra of $G_q$-equivariant operators on the space of functions $k[G_q/B_q]$. This algebra is in fact the Hecke algebra.

It is a matter of interest to interpret the double cosets directly as operators on $k[G_q/B_q]$, and in particular the cosets $B s_i B$ where $s_i$ is a Coxeter generator.

To be continued\ldots{}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[double Hecke algebra]]


\item [[double affine Hecke algebra]]


\item [[Hecke correspondence]]


\item [[Hecke category]]


\item [[Iwahori–Hecke algebra]]



\end{itemize}
\hypertarget{references_and_links}{}\subsection*{{References and links}}\label{references_and_links}

\begin{itemize}%
\item Secret Blogging Seminar: \href{https://sbseminar.wordpress.com/2008/02/25/interpreting-the-hecke-algebra/}{Interpreting the Hecke algebra}


\item Secret Blogging Seminar: \href{https://sbseminar.wordpress.com/2009/04/09/interpreting-the-hecke-algebra-ii-the-sheafification/}{Interpreting the Hecke Algebra II: the sheafification}



\end{itemize}
For the representation theory of the degenerate affine Hecke algebra see

\begin{itemize}%
\item Takeshi Suzuki, \emph{Rogawski's conjecture on the Jantzen filtration for the degenerate affine Hecke algebra of type A}, \href{http://arxiv.org/abs/math/9805035}{math.QA/9805035}

\end{itemize}


\end{document}