$rho$-shift in parabolically induced representations
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In order to define the principal series representations one takes an irreducible, unitary representation $sigma$ of $M$ (here $G$ is a semisimple Lie group with Iwasawa decomposition $G=KAN$ and $M$ is the centralizer of $A$ in $K$) and a character $nuinmathfrak{a'}_mathbb{C}$ to obtain an $MA$-representation. One extends trivially to get an $MAN$-module. Now one induces this representation to $G$. However the induced space consists of functions $f$ that satisfy $f(kman)=e^{-(nu+rho)(log a)}sigma(m)^{-1}f(k)$.
Why is there a $rho$-shift and for which reason does one need it?
representation-theory lie-groups
$endgroup$
add a comment |
$begingroup$
In order to define the principal series representations one takes an irreducible, unitary representation $sigma$ of $M$ (here $G$ is a semisimple Lie group with Iwasawa decomposition $G=KAN$ and $M$ is the centralizer of $A$ in $K$) and a character $nuinmathfrak{a'}_mathbb{C}$ to obtain an $MA$-representation. One extends trivially to get an $MAN$-module. Now one induces this representation to $G$. However the induced space consists of functions $f$ that satisfy $f(kman)=e^{-(nu+rho)(log a)}sigma(m)^{-1}f(k)$.
Why is there a $rho$-shift and for which reason does one need it?
representation-theory lie-groups
$endgroup$
add a comment |
$begingroup$
In order to define the principal series representations one takes an irreducible, unitary representation $sigma$ of $M$ (here $G$ is a semisimple Lie group with Iwasawa decomposition $G=KAN$ and $M$ is the centralizer of $A$ in $K$) and a character $nuinmathfrak{a'}_mathbb{C}$ to obtain an $MA$-representation. One extends trivially to get an $MAN$-module. Now one induces this representation to $G$. However the induced space consists of functions $f$ that satisfy $f(kman)=e^{-(nu+rho)(log a)}sigma(m)^{-1}f(k)$.
Why is there a $rho$-shift and for which reason does one need it?
representation-theory lie-groups
$endgroup$
In order to define the principal series representations one takes an irreducible, unitary representation $sigma$ of $M$ (here $G$ is a semisimple Lie group with Iwasawa decomposition $G=KAN$ and $M$ is the centralizer of $A$ in $K$) and a character $nuinmathfrak{a'}_mathbb{C}$ to obtain an $MA$-representation. One extends trivially to get an $MAN$-module. Now one induces this representation to $G$. However the induced space consists of functions $f$ that satisfy $f(kman)=e^{-(nu+rho)(log a)}sigma(m)^{-1}f(k)$.
Why is there a $rho$-shift and for which reason does one need it?
representation-theory lie-groups
representation-theory lie-groups
asked Dec 2 '18 at 17:10
Luke MathwalkerLuke Mathwalker
352212
352212
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