Mixing
Indecomposable irreducible representation of $mathrm{GL}(2, F)$ over local fields
$begingroup$
Let $F$ be a local field and $pi:F^{times}=mathrm{GL}(1, F)to mathrm{GL}(2, mathbb{C})$ be indecomposable non-irreducible admissible (smooth) representation. Here indecomposable mean that it can't be represented as a direct sum of irreducible representations (i.e. not semisimple). Assume that there exists 1-dimensional invariant subspace of $mathbb{C}^{2}$. How can we show that such representation is isomorphic to $rhootimes chi$, where
$$
rho(a) = begin{pmatrix} 1 & log |a| \ 0 & 1end{pmatrix}
$$
and $chi:F^{times} to mathbb{C}^{times}$ a quasicharacter?
By the existence of 1-dimensional space, there exists a basis ${v_{1}, v_{2}}$ of $mathbb{C}^{2}$ such that $pi(a)$ can be written as
$$
pi(a) = begin{pmatrix} chi_{1}(a) & f(a) \ 0 & chi_{2}(a) end{pmatrix}.
$$
Then $pi(a)pi(b) = pi(ab) = pi(b)pi(a)$ implies that $chi_{1}, chi_{2}:F^{times} to mathbb{C}^{times}$, and
$$
f(ab) = chi_{1}(a)f(b) + chi_{2}(b)f(a) = chi_{2}(a)f(b) + chi_{1}(b)f(a).
$$
From the last equation, we have $(chi_{1}^{-1}(a)-chi_{2}^{-1}(a))f(a) = (chi_{1}^{-1}(b)-chi_{2}^{-1}(b))f(b) = (chi_{1}^{-1}(1) - chi_{2}^{-1}(1))f(1)=0$, i.e. $(chi_{1}(a) - chi_{2}(a))f(a) =0$ for all $ain F^{times}$.
I think we may show that $chi_{1} = chi_{2}$ from this, but I can't figure it out now. If we prove $chi_{1} = chi_{2}$, then $g(a):= chi_{1}^{-1}(a)f(a)$ satisfies $g(ab)=g(a) + g(b)$, so that $g$ might be a constant multiple of a log function.
representation-theory local-field
asked Jan 26 at 17:03
Seewoo LeeSeewoo Lee
7,124927
$endgroup$
add a comment |
$begingroup$
Let $F$ be a local field and $pi:F^{times}=mathrm{GL}(1, F)to mathrm{GL}(2, mathbb{C})$ be indecomposable non-irreducible admissible (smooth) representation. Here indecomposable mean that it can't be represented as a direct sum of irreducible representations (i.e. not semisimple). Assume that there exists 1-dimensional invariant subspace of $mathbb{C}^{2}$. How can we show that such representation is isomorphic to $rhootimes chi$, where
$$
rho(a) = begin{pmatrix} 1 & log |a| \ 0 & 1end{pmatrix}
$$
and $chi:F^{times} to mathbb{C}^{times}$ a quasicharacter?
By the existence of 1-dimensional space, there exists a basis ${v_{1}, v_{2}}$ of $mathbb{C}^{2}$ such that $pi(a)$ can be written as
$$
pi(a) = begin{pmatrix} chi_{1}(a) & f(a) \ 0 & chi_{2}(a) end{pmatrix}.
$$
Then $pi(a)pi(b) = pi(ab) = pi(b)pi(a)$ implies that $chi_{1}, chi_{2}:F^{times} to mathbb{C}^{times}$, and
$$
f(ab) = chi_{1}(a)f(b) + chi_{2}(b)f(a) = chi_{2}(a)f(b) + chi_{1}(b)f(a).
$$
From the last equation, we have $(chi_{1}^{-1}(a)-chi_{2}^{-1}(a))f(a) = (chi_{1}^{-1}(b)-chi_{2}^{-1}(b))f(b) = (chi_{1}^{-1}(1) - chi_{2}^{-1}(1))f(1)=0$, i.e. $(chi_{1}(a) - chi_{2}(a))f(a) =0$ for all $ain F^{times}$.
I think we may show that $chi_{1} = chi_{2}$ from this, but I can't figure it out now. If we prove $chi_{1} = chi_{2}$, then $g(a):= chi_{1}^{-1}(a)f(a)$ satisfies $g(ab)=g(a) + g(b)$, so that $g$ might be a constant multiple of a log function.
representation-theory local-field
asked Jan 26 at 17:03
Seewoo LeeSeewoo Lee
7,124927
$endgroup$
add a comment |
$begingroup$
Let $F$ be a local field and $pi:F^{times}=mathrm{GL}(1, F)to mathrm{GL}(2, mathbb{C})$ be indecomposable non-irreducible admissible (smooth) representation. Here indecomposable mean that it can't be represented as a direct sum of irreducible representations (i.e. not semisimple). Assume that there exists 1-dimensional invariant subspace of $mathbb{C}^{2}$. How can we show that such representation is isomorphic to $rhootimes chi$, where
$$
rho(a) = begin{pmatrix} 1 & log |a| \ 0 & 1end{pmatrix}
$$
and $chi:F^{times} to mathbb{C}^{times}$ a quasicharacter?
By the existence of 1-dimensional space, there exists a basis ${v_{1}, v_{2}}$ of $mathbb{C}^{2}$ such that $pi(a)$ can be written as
$$
pi(a) = begin{pmatrix} chi_{1}(a) & f(a) \ 0 & chi_{2}(a) end{pmatrix}.
$$
Then $pi(a)pi(b) = pi(ab) = pi(b)pi(a)$ implies that $chi_{1}, chi_{2}:F^{times} to mathbb{C}^{times}$, and
$$
f(ab) = chi_{1}(a)f(b) + chi_{2}(b)f(a) = chi_{2}(a)f(b) + chi_{1}(b)f(a).
$$
From the last equation, we have $(chi_{1}^{-1}(a)-chi_{2}^{-1}(a))f(a) = (chi_{1}^{-1}(b)-chi_{2}^{-1}(b))f(b) = (chi_{1}^{-1}(1) - chi_{2}^{-1}(1))f(1)=0$, i.e. $(chi_{1}(a) - chi_{2}(a))f(a) =0$ for all $ain F^{times}$.
I think we may show that $chi_{1} = chi_{2}$ from this, but I can't figure it out now. If we prove $chi_{1} = chi_{2}$, then $g(a):= chi_{1}^{-1}(a)f(a)$ satisfies $g(ab)=g(a) + g(b)$, so that $g$ might be a constant multiple of a log function.
representation-theory local-field
asked Jan 26 at 17:03
Seewoo LeeSeewoo Lee
7,124927
$endgroup$
Let $F$ be a local field and $pi:F^{times}=mathrm{GL}(1, F)to mathrm{GL}(2, mathbb{C})$ be indecomposable non-irreducible admissible (smooth) representation. Here indecomposable mean that it can't be represented as a direct sum of irreducible representations (i.e. not semisimple). Assume that there exists 1-dimensional invariant subspace of $mathbb{C}^{2}$. How can we show that such representation is isomorphic to $rhootimes chi$, where
$$
rho(a) = begin{pmatrix} 1 & log |a| \ 0 & 1end{pmatrix}
$$
and $chi:F^{times} to mathbb{C}^{times}$ a quasicharacter?
By the existence of 1-dimensional space, there exists a basis ${v_{1}, v_{2}}$ of $mathbb{C}^{2}$ such that $pi(a)$ can be written as
$$
pi(a) = begin{pmatrix} chi_{1}(a) & f(a) \ 0 & chi_{2}(a) end{pmatrix}.
$$
Then $pi(a)pi(b) = pi(ab) = pi(b)pi(a)$ implies that $chi_{1}, chi_{2}:F^{times} to mathbb{C}^{times}$, and
$$
f(ab) = chi_{1}(a)f(b) + chi_{2}(b)f(a) = chi_{2}(a)f(b) + chi_{1}(b)f(a).
$$
From the last equation, we have $(chi_{1}^{-1}(a)-chi_{2}^{-1}(a))f(a) = (chi_{1}^{-1}(b)-chi_{2}^{-1}(b))f(b) = (chi_{1}^{-1}(1) - chi_{2}^{-1}(1))f(1)=0$, i.e. $(chi_{1}(a) - chi_{2}(a))f(a) =0$ for all $ain F^{times}$.
I think we may show that $chi_{1} = chi_{2}$ from this, but I can't figure it out now. If we prove $chi_{1} = chi_{2}$, then $g(a):= chi_{1}^{-1}(a)f(a)$ satisfies $g(ab)=g(a) + g(b)$, so that $g$ might be a constant multiple of a log function.
representation-theory local-field
representation-theory local-field
asked Jan 26 at 17:03
Seewoo LeeSeewoo Lee
7,124927
asked Jan 26 at 17:03
Seewoo LeeSeewoo Lee
7,124927
asked Jan 26 at 17:03
Seewoo LeeSeewoo Lee
7,124927
asked Jan 26 at 17:03
Seewoo LeeSeewoo Lee
7,124927
asked Jan 26 at 17:03
Seewoo LeeSeewoo Lee
7,124927
7,124927
add a comment |
add a comment |
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