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$begingroup$

I have a question about a small point in Vigneres' introductory notes on the trace formula.

Here $G$ is a finite group, $L$ is an algebraically closed field (if necessary assume $operatornamecharL$ does not divide $|G|$), and $L(G)$ is the space of functions $G rightarrow L$, which is a representation $R$ of $G times G$ under $R(x,y)f(g) = f(x^-1gy)$. The set of irreducible representations over $L$ of $G$ is denoted $S_L$.

If $(sigma,W)$ is a representation of $G$, and $f in L(G)$, then $sigma(f) in operatornameEnd(W)$ is defined by

$$sigma(f)w = sumlimits_x in G f(x) sigma(x)w$$

I don't understand how Schur's lemma is being used in this section:

Let $(V,pi)$ and $(W,sigma)$ be two irreducible representations of $G$. I guess they are considering the composition

$$sigma circ phi_pi: operatornameEnd_L(V^ast) rightarrow operatornameEnd_L(W)$$

Maybe I am not understanding the notation correctly. Taken literally it's saying this map sends everything in $operatornameEnd_L(V)$ to a constant multiple $lambda(pi)$ of the identity of $V$, and the constant only depends on $pi$.

representation-theory

share|cite|improve this question

edited Mar 13 at 19:20

D_S

asked Mar 13 at 18:34

D_SD_S

14k61553

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Okay I think I see what's going on. The given map $sigma circ phi_pi$ is an intertwining operator $$pi^ast otimes pi rightarrow sigma otimes sigma^ast$$ between irreducible $G times G$-representations. These are isomorphic if and only if $pi cong sigma^ast$. If so, then it sends $1_V^ast$ to a scalar multiple of $1_W$ by Schur's lemma. If not, then the map has to be zero.
  $endgroup$
  – D_S
  Mar 13 at 19:23
  
  

  
  
  
  

add a comment | 

0

$begingroup$

I have a question about a small point in Vigneres' introductory notes on the trace formula.

Here $G$ is a finite group, $L$ is an algebraically closed field (if necessary assume $operatornamecharL$ does not divide $|G|$), and $L(G)$ is the space of functions $G rightarrow L$, which is a representation $R$ of $G times G$ under $R(x,y)f(g) = f(x^-1gy)$. The set of irreducible representations over $L$ of $G$ is denoted $S_L$.

If $(sigma,W)$ is a representation of $G$, and $f in L(G)$, then $sigma(f) in operatornameEnd(W)$ is defined by

$$sigma(f)w = sumlimits_x in G f(x) sigma(x)w$$

I don't understand how Schur's lemma is being used in this section:

Let $(V,pi)$ and $(W,sigma)$ be two irreducible representations of $G$. I guess they are considering the composition

$$sigma circ phi_pi: operatornameEnd_L(V^ast) rightarrow operatornameEnd_L(W)$$

Maybe I am not understanding the notation correctly. Taken literally it's saying this map sends everything in $operatornameEnd_L(V)$ to a constant multiple $lambda(pi)$ of the identity of $V$, and the constant only depends on $pi$.

representation-theory

share|cite|improve this question

edited Mar 13 at 19:20

D_S

asked Mar 13 at 18:34

D_SD_S

14k61553

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Okay I think I see what's going on. The given map $sigma circ phi_pi$ is an intertwining operator $$pi^ast otimes pi rightarrow sigma otimes sigma^ast$$ between irreducible $G times G$-representations. These are isomorphic if and only if $pi cong sigma^ast$. If so, then it sends $1_V^ast$ to a scalar multiple of $1_W$ by Schur's lemma. If not, then the map has to be zero.
  $endgroup$
  – D_S
  Mar 13 at 19:23
  
  

  
  
  
  

add a comment | 

$begingroup$

I have a question about a small point in Vigneres' introductory notes on the trace formula.

Here $G$ is a finite group, $L$ is an algebraically closed field (if necessary assume $operatornamecharL$ does not divide $|G|$), and $L(G)$ is the space of functions $G rightarrow L$, which is a representation $R$ of $G times G$ under $R(x,y)f(g) = f(x^-1gy)$. The set of irreducible representations over $L$ of $G$ is denoted $S_L$.

If $(sigma,W)$ is a representation of $G$, and $f in L(G)$, then $sigma(f) in operatornameEnd(W)$ is defined by

$$sigma(f)w = sumlimits_x in G f(x) sigma(x)w$$

I don't understand how Schur's lemma is being used in this section:

Let $(V,pi)$ and $(W,sigma)$ be two irreducible representations of $G$. I guess they are considering the composition

$$sigma circ phi_pi: operatornameEnd_L(V^ast) rightarrow operatornameEnd_L(W)$$

Maybe I am not understanding the notation correctly. Taken literally it's saying this map sends everything in $operatornameEnd_L(V)$ to a constant multiple $lambda(pi)$ of the identity of $V$, and the constant only depends on $pi$.

representation-theory

share|cite|improve this question

edited Mar 13 at 19:20

D_S

asked Mar 13 at 18:34

D_SD_S

14k61553

$endgroup$

share|cite|improve this question

edited Mar 13 at 19:20

D_S

asked Mar 13 at 18:34

D_SD_S

14k61553

share|cite|improve this question

edited Mar 13 at 19:20

D_S

asked Mar 13 at 18:34

D_SD_S

14k61553

asked Mar 13 at 18:34

D_SD_S

14k61553

asked Mar 13 at 18:34

D_SD_S

14k61553