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Classification of (finite dimensional) admissible representations of $F^times = mathrmGL(1, F)$.

Showing that a given representation is not completely reducibleDecomposition of symmetric powers of $mathrmsl_2$ representationsClassification of irreducible (g,K)-modules for other g than sl2Irreps of products between dihedral group and any finite groupDeterminant of a characterPrerequisites for learning about lie algebras and their representations.Existence of conductor for admissible representations of $GL_2(mathbbQ_p)$Classification of $(mathfrakg,K)$-module of $mathrmGL(2, mathbbR)$Whittaker model for $mathrmGL(2, mathbbR)$Test vector for local zeta integral with ramified character

3

$begingroup$

Let $F$ be a non-archimedean local field. In Bump's textbook, there are two kinds of 2-dimensional such representations:
$$
tmapsto beginpmatrix xi(t) & \ & xi'(t)endpmatrix
$$
for quasi-characters $xi, xi':F^times to mathbbC^times$, or
$$
tmapsto xi(t) beginpmatrix 1& v(t)\&1endpmatrix
$$
for some quasi-character $xi$ and a valuation map $v:F^times to mathbbZ$.

I want to know the complete classification of every (finite dimensional) admissible representation of $F^times$ (including reducible ones). It seems that the above representations like Jordan blocks, so maybe all the finite dimensional representations look like the above ones. Is this true?

number-theory representation-theory p-adic-number-theory automorphic-forms

share|cite|improve this question

asked Mar 28 at 20:35

Seewoo LeeSeewoo Lee

7,0851929

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  With $F = BbbQ_p$ you can look at $rho(p), rho(zeta_p-1), rho(1+p)$ and (multiplying by a character $F^times to BbbC^times$) assume they are of determinant $1$, in some basis $rho(zeta_p-1),rho(1+p)$ are diagonal, $rho(zeta_p-1)^p-1 = I, rho(1+p)^p^m = I$ and $rho(p)$ is upper triangular and commutes with them
  $endgroup$
  – reuns
  Mar 28 at 23:47
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  $F^times$ is abelian, so irreducible representations are just quasicharacters, which are one-dimensional.
  $endgroup$
  – Peter Humphries
  Mar 29 at 11:33
  

  
  
  
  

add a comment | 

3

$begingroup$

Let $F$ be a non-archimedean local field. In Bump's textbook, there are two kinds of 2-dimensional such representations:
$$
tmapsto beginpmatrix xi(t) & \ & xi'(t)endpmatrix
$$
for quasi-characters $xi, xi':F^times to mathbbC^times$, or
$$
tmapsto xi(t) beginpmatrix 1& v(t)\&1endpmatrix
$$
for some quasi-character $xi$ and a valuation map $v:F^times to mathbbZ$.

I want to know the complete classification of every (finite dimensional) admissible representation of $F^times$ (including reducible ones). It seems that the above representations like Jordan blocks, so maybe all the finite dimensional representations look like the above ones. Is this true?

number-theory representation-theory p-adic-number-theory automorphic-forms

share|cite|improve this question

asked Mar 28 at 20:35

Seewoo LeeSeewoo Lee

7,0851929

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  With $F = BbbQ_p$ you can look at $rho(p), rho(zeta_p-1), rho(1+p)$ and (multiplying by a character $F^times to BbbC^times$) assume they are of determinant $1$, in some basis $rho(zeta_p-1),rho(1+p)$ are diagonal, $rho(zeta_p-1)^p-1 = I, rho(1+p)^p^m = I$ and $rho(p)$ is upper triangular and commutes with them
  $endgroup$
  – reuns
  Mar 28 at 23:47
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  $F^times$ is abelian, so irreducible representations are just quasicharacters, which are one-dimensional.
  $endgroup$
  – Peter Humphries
  Mar 29 at 11:33
  

  
  
  
  

add a comment | 

$begingroup$

Let $F$ be a non-archimedean local field. In Bump's textbook, there are two kinds of 2-dimensional such representations:
$$
tmapsto beginpmatrix xi(t) & \ & xi'(t)endpmatrix
$$
for quasi-characters $xi, xi':F^times to mathbbC^times$, or
$$
tmapsto xi(t) beginpmatrix 1& v(t)\&1endpmatrix
$$
for some quasi-character $xi$ and a valuation map $v:F^times to mathbbZ$.

I want to know the complete classification of every (finite dimensional) admissible representation of $F^times$ (including reducible ones). It seems that the above representations like Jordan blocks, so maybe all the finite dimensional representations look like the above ones. Is this true?

number-theory representation-theory p-adic-number-theory automorphic-forms

share|cite|improve this question

asked Mar 28 at 20:35

Seewoo LeeSeewoo Lee

7,0851929

$endgroup$

share|cite|improve this question

asked Mar 28 at 20:35

Seewoo LeeSeewoo Lee

7,0851929

share|cite|improve this question

asked Mar 28 at 20:35

Seewoo LeeSeewoo Lee

7,0851929

asked Mar 28 at 20:35

Seewoo LeeSeewoo Lee

7,0851929

asked Mar 28 at 20:35

Seewoo LeeSeewoo Lee

7,0851929