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Deciding whether a representation is orthogonal or symplectic

irreducible highest weight modulesRepresentations of non-semisimple Lie algebrasWeight spaces of Verma modulesThe relation between Weyl character formula and Frobenius characteristic mapBasis for complex representation with symmetric bilinear formEvery submodule of a cyclic $mathfrakg$-module is a weight moduleCriterion for Checking When a Lie Algebra Module is IrreducibleWeight $mathfraksl_2$-module with finite dimensional weight spaces has finite length?Understanding weight spaces of weight module from its composition factors?Irreducible Dual Representation

2

$begingroup$

I'm trying to understand the proof of Proposition 7 part (iii) from this paper of Dadok https://www.ams.org/journals/tran/1985-288-01/S0002-9947-1985-0773051-1/S0002-9947-1985-0773051-1.pdf (Statement of the proposition on page 128, proof on page 131), but getting stuck on one line in the the middle.

The set-up is as follows: We have an irreducible self-dual (complex) representation $pi_lambda:mathfrakgto mathfrakgl(V)$ of highest weight $lambda$ (say $v_lambdain V$ is a heighest weight vector). There is a distinguished set $beta_1,dots, beta_l$ of strongly orthogonal positive roots of $mathfrakg$ (their specific definition isn't important for this part), and we are assuming that $lambda$ is in the (real) span of the $beta_i$. We are considering the subalgebra $mathfrakusubseteq mathfrakg$ generated by the rootspaces of the $pm beta_i$ (note $mathfrakucong mathfraksl_2(mathbbC)^oplus l$).

Self-duality of $pi_lambda$ implies that there is a non-degenerate invariant bilinear form $B:Vtimes Vto mathbbC$, and to determine whether $pi_lambda$ is orthogonal or symplectic we need to figure out whether $B$ is symmetric or skew-symmetric. Now he defines $U_lambdasubseteq V_lambda$ as the $U(mathfraku)$-submodule generated by the highest weight vector $v_lambda$ (the module $U_lambda$ will be simple by the theorem of highest weight, using the fact that $lambda$ is in the span of the $beta_i$). The idea of the argument is that he wants to show that one decide whether $B$ is symmetric or skew-symmetric by looking at the restriction $B:U_lambdatimes U_lambdato mathbbC$. To that end, he writes:

"If $B:Vto Vto mathbbC$ is the nondegenerate bilinear form invariant under $pi_lambda$, we see that $B$ must remain non-degenerate on $U_lambdatimes U_lambdatomathbbC$ since the $mathfraku$-module $U_lambda$ appears in $V_lambda$ with multiplicity one."

Can anyone help me make sense of this line? I can't even figure out what he means by multiplicity in this context.

Thanks!

abstract-algebra modules representation-theory lie-algebras

share|cite|improve this question

edited Mar 22 at 19:42

itinerantleopard

asked Mar 21 at 14:53

itinerantleoparditinerantleopard

1379

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Why is $mathfrakucongmathfraksl_2$? Isn't it generated by multiple root spaces, or just $pmbeta_i$ for some $i$?
  $endgroup$
  – David Hill
  Mar 22 at 15:50
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Ah, yes thank you, that was a typo! It should have said $ucong mathfraksl_2(mathbbC)^oplus l$.
  $endgroup$
  – itinerantleopard
  Mar 22 at 19:41
  

  
  
  
  

add a comment | 

2

$begingroup$

I'm trying to understand the proof of Proposition 7 part (iii) from this paper of Dadok https://www.ams.org/journals/tran/1985-288-01/S0002-9947-1985-0773051-1/S0002-9947-1985-0773051-1.pdf (Statement of the proposition on page 128, proof on page 131), but getting stuck on one line in the the middle.

The set-up is as follows: We have an irreducible self-dual (complex) representation $pi_lambda:mathfrakgto mathfrakgl(V)$ of highest weight $lambda$ (say $v_lambdain V$ is a heighest weight vector). There is a distinguished set $beta_1,dots, beta_l$ of strongly orthogonal positive roots of $mathfrakg$ (their specific definition isn't important for this part), and we are assuming that $lambda$ is in the (real) span of the $beta_i$. We are considering the subalgebra $mathfrakusubseteq mathfrakg$ generated by the rootspaces of the $pm beta_i$ (note $mathfrakucong mathfraksl_2(mathbbC)^oplus l$).

Self-duality of $pi_lambda$ implies that there is a non-degenerate invariant bilinear form $B:Vtimes Vto mathbbC$, and to determine whether $pi_lambda$ is orthogonal or symplectic we need to figure out whether $B$ is symmetric or skew-symmetric. Now he defines $U_lambdasubseteq V_lambda$ as the $U(mathfraku)$-submodule generated by the highest weight vector $v_lambda$ (the module $U_lambda$ will be simple by the theorem of highest weight, using the fact that $lambda$ is in the span of the $beta_i$). The idea of the argument is that he wants to show that one decide whether $B$ is symmetric or skew-symmetric by looking at the restriction $B:U_lambdatimes U_lambdato mathbbC$. To that end, he writes:

"If $B:Vto Vto mathbbC$ is the nondegenerate bilinear form invariant under $pi_lambda$, we see that $B$ must remain non-degenerate on $U_lambdatimes U_lambdatomathbbC$ since the $mathfraku$-module $U_lambda$ appears in $V_lambda$ with multiplicity one."

Can anyone help me make sense of this line? I can't even figure out what he means by multiplicity in this context.

Thanks!

abstract-algebra modules representation-theory lie-algebras

share|cite|improve this question

edited Mar 22 at 19:42

itinerantleopard

asked Mar 21 at 14:53

itinerantleoparditinerantleopard

1379

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Why is $mathfrakucongmathfraksl_2$? Isn't it generated by multiple root spaces, or just $pmbeta_i$ for some $i$?
  $endgroup$
  – David Hill
  Mar 22 at 15:50
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Ah, yes thank you, that was a typo! It should have said $ucong mathfraksl_2(mathbbC)^oplus l$.
  $endgroup$
  – itinerantleopard
  Mar 22 at 19:41
  

  
  
  
  

add a comment | 

$begingroup$

I'm trying to understand the proof of Proposition 7 part (iii) from this paper of Dadok https://www.ams.org/journals/tran/1985-288-01/S0002-9947-1985-0773051-1/S0002-9947-1985-0773051-1.pdf (Statement of the proposition on page 128, proof on page 131), but getting stuck on one line in the the middle.

The set-up is as follows: We have an irreducible self-dual (complex) representation $pi_lambda:mathfrakgto mathfrakgl(V)$ of highest weight $lambda$ (say $v_lambdain V$ is a heighest weight vector). There is a distinguished set $beta_1,dots, beta_l$ of strongly orthogonal positive roots of $mathfrakg$ (their specific definition isn't important for this part), and we are assuming that $lambda$ is in the (real) span of the $beta_i$. We are considering the subalgebra $mathfrakusubseteq mathfrakg$ generated by the rootspaces of the $pm beta_i$ (note $mathfrakucong mathfraksl_2(mathbbC)^oplus l$).

Self-duality of $pi_lambda$ implies that there is a non-degenerate invariant bilinear form $B:Vtimes Vto mathbbC$, and to determine whether $pi_lambda$ is orthogonal or symplectic we need to figure out whether $B$ is symmetric or skew-symmetric. Now he defines $U_lambdasubseteq V_lambda$ as the $U(mathfraku)$-submodule generated by the highest weight vector $v_lambda$ (the module $U_lambda$ will be simple by the theorem of highest weight, using the fact that $lambda$ is in the span of the $beta_i$). The idea of the argument is that he wants to show that one decide whether $B$ is symmetric or skew-symmetric by looking at the restriction $B:U_lambdatimes U_lambdato mathbbC$. To that end, he writes:

"If $B:Vto Vto mathbbC$ is the nondegenerate bilinear form invariant under $pi_lambda$, we see that $B$ must remain non-degenerate on $U_lambdatimes U_lambdatomathbbC$ since the $mathfraku$-module $U_lambda$ appears in $V_lambda$ with multiplicity one."

Can anyone help me make sense of this line? I can't even figure out what he means by multiplicity in this context.

Thanks!

abstract-algebra modules representation-theory lie-algebras

share|cite|improve this question

edited Mar 22 at 19:42

itinerantleopard

asked Mar 21 at 14:53

itinerantleoparditinerantleopard

1379

$endgroup$

share|cite|improve this question

edited Mar 22 at 19:42

itinerantleopard

asked Mar 21 at 14:53

itinerantleoparditinerantleopard

1379

share|cite|improve this question

edited Mar 22 at 19:42

itinerantleopard

asked Mar 21 at 14:53

itinerantleoparditinerantleopard

1379

asked Mar 21 at 14:53

itinerantleoparditinerantleopard

1379

asked Mar 21 at 14:53

itinerantleoparditinerantleopard

1379