Etingof proposition 3.1.4

Question

I am reading Etingof et al's Introduction to Representation Theory.

First they prove that any irreducible subrepresentation is isomorphic to one of the $V_i$s by Schur's lemma. Further the inclusion of this irreducible after identification with $V_i$ will look like:

I do not quite understand what they do after this: $G_i$ acts on $n_iV_i$ and therefore on all of $V$. This I understand. But why does it act on the set of subrepresentations of $V$? In the proof what are the matrices $X_i$? If someone could elaborate on these first then I will be able to give another go at the proof.

I can see that what we want to say is subrepresentation of semisimple is semisimple. I can prove it in other ways. And the matrices $X_i$ can be found by using Schur's lemma and 'Hom and direct sum commutes' property. But I can't follow the argument of Etingof.

Answer 1

The group $G_i=GL_{n_i}(k)$ acts on the set of subrepresentations of $V$.

This implies that for a subrepresentation $W$ of $V$ then $Wg_i$ is also a subrepresentation of $V$.

Proof. The only nontrivial part is to check that each element $g_i$ in $G_i$ sends a subrepresentation $W$ of $V$ to another subrepresentation $Wg_i$ of $V$. Indeed, fix $a\in A$ then $aw=w'\in W$ for any $w\in W$. We show $a(wg_i)=w'g_i$.

We first seperate $w,w'$ into $w=(w_1,w_2), w'=(w_1',w_2')$ where $w_1,w_1'\in \bigoplus_{j\ne i} n_jV_j$ and $w_2,w_2'\in n_iV_i$. Note $wg_i=(w_1,w_2g_i)$ so $a(wg_i)=w'g_i$ is equivalent to $a(w_1,w_2g_i)=(w_1',w_2'g_i)$. Since $\bigoplus_{j\ne i} n_jV_j$ and $n_iV_i$ are subrepresentations of $V$ so their direct sum is a representation with action defined as $a(w_1,w_2g_i)=(aw_1,a(w_2g_i))$. From this, we find $a(wg_i)=w'g_i$ is equivalent to $aw_1=w_1',a(w_2)g_i=w_2'g_i$. This holds since $aw=w'$ or $(aw_1,aw_2)=(w_1',w_2')$.

The check that this is indeed a group action is not hard.

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Etingof's proof.

• First show $W$ has an irreducible subrepresentation $P$ isomorphic to $V_i$ via identification $v\mapsto (vq_1,\ldots, vq_{n_i})$ for $v\in V_i,q_j\in k$.
• Choose $g_i\in G_i$ so $(q_1,\ldots, q_{n_i})g_i=(1,0,\ldots,0)$ then $Wg_i$ has irreducible subrepresentation $Pg_i$, which is the first summand $V_i$ of $nV_i$. Take the projection $Wg_i\to V_i$ then with $W'=\ker(Wg_i\to V_i)$, we have $Wg_i=V_i\oplus W'$. Furthermore, $W'\subset n_1V_1 \oplus \cdots \oplus (n_i-1)V_i \oplus n_mV_m$ so by inductive hypothesis, $W'$ is semisimple with inclusion map as defined. From this, we find $Wg_i$ is semisimple and has the inclusion map $Wg_i\to V$ as described.
• Coming back to the action of $G_i$ on set of subrepresentations of $V$, this action preserves the property that if subrepresentation $W$ of $V$ is semisimple has inclusion map as described then $Wg_i$ is also semisimple and the inclusion map $Wg_i$ also has that property (but under different choice of matrix $X_i$ of course).
• So now we know $Wg_i$ is semisimple and has inclusion map as described. From previous argument, $W=(Wg_i)g_i^{-1}$ is semisimple and has inclusion map as described.

So in Etingof's proof, he didn't specifically describe the matrix $X_i$. He only mentioned that the inclusion $W\to V$ is described by some matrices' $X_i$'s.