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In the book "Representation of Semisimple Lie Algebra in the BGG Category $\mathcal{O}$".

Exercise 1.13. Suppose $\lambda\not\in\Lambda$, so the linkage class $W\cdot\lambda$ is the disjoint union of its nonempty intersections with various cosets of $\Lambda_r$ in $\mathbb{h}^*$. Prove that each $M\in\mathcal{O}_{\chi_\lambda}$ has a corresponding direct sum decomposition $M=\oplus_i M_i$, in which all weights of $M_i$ lies in a single coset.

My attempt:

$M=\bigoplus_{\nu\in\mathfrak{h}^*} M_\nu =\bigoplus_{[\nu]\in\mathfrak{h}^*/\Lambda_r} M^{[\nu]}$.

Since $\mathfrak{g}_\alpha\cdot M_\mu\subseteq M_{\mu+\alpha}$ for all $\alpha\in \Phi$, we get $U(\mathfrak{n})\cdot M^{[\nu]}\subseteq M^{[\nu]}$, $U(\mathfrak{h})\cdot M^{[\nu]}\subseteq M^{[\nu]}$ and $U(\mathfrak{n}^-)\cdot M^{[\nu]}\subseteq M^{[\nu]}$.

Since $U(\mathfrak{g})=U(\mathfrak{n}^-)U(\mathfrak{h})U(\mathfrak{n})$, we get $U(\mathfrak{g})\cdot M^{[\nu]}\subseteq M^{[\nu]}$. Hence $M^{[\nu]}$ is a $U(\mathfrak{g})$-submodule of $M$.

Since $M\in\mathcal{O}$, $M$ is finitely generated as a $U(\mathfrak{g})$-module. Therefore, $M=\bigoplus_{i=1}^n M^{[\nu_i]}$.

Now, let $W\cdot\lambda=\{\eta_1,\cdots,\eta_k\}$. Consider $\{\eta_{i_1},\cdots, \eta_{i_r}\}\subseteq \{\eta_1,\cdots,\eta_k\}$ such that $[\eta_{i_1}],\cdots, [\eta_{i_r}]$ are distinct and $\{[\eta_{i_1}],\cdots, [\eta_{i_r}]\}=\{[\eta_1],\cdots,[\eta_k]\}$. It is clearly that $W\cdot\lambda\cap[\eta]\neq \emptyset\implies [\eta]\in \{ [\eta_{i_1}],\cdots, [\eta_{i_r}]\}$.

Then $W\cdot \lambda =\bigcup_{\eta\in\mathfrak{h}^*} W\cdot\lambda \cap[\eta] =\bigsqcup_{j=1}^{r}W\cdot\lambda \cap[\eta_{i_j}] =\bigsqcup_{j=1}^{r}W\cdot \eta_{i_j} \cap[\eta_{i_j}] =\bigsqcup_{j=1}^{r}W_{[\eta_{i_j}]}\cdot \eta_{i_j}$

I would like to know whether the corresponding direct sum decomposition means $M=\bigoplus_{j=1}^r M^{[\eta_{i_j}]}$. If so, how to prove it? Also, am I on the right track?

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I think you're overcomplicating things. You have showed that: (a) As a vector space we have the following decomposition: $$ M = \bigoplus_{[\nu]\in\mathfrak{h}^*/\Lambda_r} M^{[\nu]}. $$ (b) Furthermore, each $M^{[\nu]}$ is a submodule of $M$.

From this it follows directly that the direct sum above is a decomposition of modules.

Edit to clarify: Note that just before Exercise 1.13, Humphreys writes "The following easy exercise gives a refinement of the linkage classes in the nonintegral case..." (emphasis mine). Chapter 1.1 has just introduced Category $\mathcal{O}$, and Exercise 1.1(b) shows that $\mathcal{O}$ decomposes as

$$\mathcal{O}=\bigoplus_{[\nu]∈\mathfrak{h}^∗/\Lambda_r}\mathcal{O}^{[\nu]}.$$

Chapter 1.12 shows that $\mathcal{O}$ also decomposes by central character as

$$\mathcal{O}=\bigoplus_{\chi}\mathcal{O}_\chi.$$

Chapter 1.13 introduces the notion of a block of a category, and shows that if $\lambda$ is integral then $\mathcal{O}_{\chi_\lambda}$ is a block. Exercise 1.13 is about what happens when $\lambda$ is not integral, and is a straightforward application of the result of Exercise 1.1(b). In this case $[w\cdot\lambda]$ is not a single coset of $\Lambda_r$ in $\mathfrak{h}^*$, but several, say $[w\cdot\lambda] = \{[\nu_1], \dots, [\nu_k]\}$. Exercise 1.1(b) now tells us that

$$\mathcal{O}_{\chi_\lambda}=\bigoplus_{i=1}^k\mathcal{O}_{\chi\lambda}^{[\nu_i]},$$

so when $\lambda$ is not integral then $\mathcal{O}_{\chi_\lambda}$ splits into several blocks, each corresponding to one of the cosets in $[w\cdot\lambda]$.

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  • $\begingroup$ I do not think I am overcomplicating thing since what you have suggested is the exercise 1.1 in Humphreys' Category $\mathcal{O}$ book. But what I am doing is exercise 1.13. I fairly sure the word "corresponding" has something to do with the decomposition of $W\cdot\lambda$ into disjoint union. $\endgroup$ – James Cheung Sep 21 '18 at 11:58
  • $\begingroup$ I've edited my answer, hopefully this clears things up? $\endgroup$ – Johan Kåhrström Sep 24 '18 at 9:06
  • $\begingroup$ Now it is much clearer but I still DO NOT understand how to get the decomposition $\mathcal{O}_{\chi_\lambda}=\bigoplus_{i=1}^k\mathcal{O}_{\chi\lambda}^{[\nu_i]}$. Since $M$ itself (NOT its composition factor) may not have weight in the the form of $w\cdot\lambda$. $\endgroup$ – James Cheung Sep 24 '18 at 15:53
  • $\begingroup$ M is assumed to be in $\mathcal{O}_{\chi_\lambda}$, so by the Harish-Chandra Theorem (b) (Humphreys section 1.10) all composition factors of $M$ are on the form $L(\mu)$ where $\mu = w\cdot \lambda$ for some $w\in W$. Is that enough? $\endgroup$ – Johan Kåhrström Sep 25 '18 at 20:59
  • $\begingroup$ By corollary 1.2 in Humphreys' book, what you have said indeed tell $M\in\mathcal{O}_{\chi_\lambda}$ and also $M^{[\nu]}\in\mathcal{O}_{\chi_\lambda}$ have weight of form $w\cdot\lambda$, for some $w\in W$. Yes. I think your proof also works as well as my proof below. $\endgroup$ – James Cheung Sep 26 '18 at 7:43
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In Humphreys' book, I quote "A formal argument (which we leave to the reader) shows that M decomposes uniquely as a direct sum of submodules, each belonging to a single block. In particular, each indecomposable module belongs to a single block. (See for example Jantzen [153, II.7.1]; his finite dimensionality assumption can be replaced here by the chain conditions on modules.)"

Remark: Now if $M$ is arbitrary, we say it belongs to a block if all its composition factors do.

The blocks of $\mathcal{O}$ are precisely the subcategories consisting of modules whose composition factors all have highest weights linked by $W_{[\lambda]}$ to an antidominant weight $\lambda$. We denote the block associated to an antidominant weight by $\mathcal{O}_\lambda$.

By mimicking the proof in Jantzen [153, II.7.1]:

Let us denote the set of blocks by $\mathcal{B}$. For any $M\in \mathcal{O}_{\chi_\lambda}$, let $M_b$ be the sum of all submodules $M'$ of $M$ such that all composition factors of $M'$ belong to $b$. Then $M_b$ is the largest submodule with this property. By the remark above, we get $M_b\in b$.

Claim $M=\bigoplus_{b\in\mathcal{B}}M_b$.

The sum is clearly direct. It suffices to show $N=\sum_{b\in\mathcal{B}}M_b$ is all of $M$.

Suppose $N\neq M$. Then $\{0\}\neq M/N\in \mathcal{O}$, by corollary 1.2. in Humphreys' book, there is a chain $\{0\}\subseteq N'/N\subseteq M/N$ such that $N'/N=L$ is simple.

Equivalently, there is a submodule $N'\subseteq M$ such that $N\subset N'$ such that $N'/N$ is simple. There is a block $b$ with $L\in b$.

Set $N''=\bigoplus_{b'\neq b}M_{b'}\subseteq N\subset N'$. Then $\{0\}\subseteq N/N''\subseteq N'/N''$.

All composition factors of $N'/N''$ belong to $b$ since they are either $L\cong (N'/N'')/(N/N'')$ or composition factors of $N/N''\cong M_b$.

By the definition of blocks, the exact sequence $0\to N''\to N'\to N'/N''\to 0$ splits. So there is a submodule $E$ of $N'$ with $N'=N''\oplus E$. We have $E\cong N'/N''$, hence $E\subseteq M_b$. It follows that $N'\subseteq \bigoplus_{b'}M_{b'}=N$, a contradiction. Therefore, the claim follows.

Now claim all weights of $M_b\in b$ lies in a single coset.

If $M_b$ is indecomposable, by exercise 1.1 (b) in Humphreys' book, we are done.

If $M_b$ is decomposable, then $M_b=M_b^1\oplus\cdots\oplus M_b^k$, where $M_b^i$ is indecomposable. Note that $M_b^i\in b$ since $M_b\in b$.

Let $b=\mathcal{O}_\lambda$. By corollary 1.2., there is a composition series for $M_b^i$: $\{0\}\subseteq M_1^i\subseteq \cdots\subseteq M_b^i$. Then $M_1^i$ have a highest weight in the form $w_i\cdot\lambda$ with $w_i\in W_{[\lambda]}$. This implies $M_b^i$ contains a weight in the form $w_i\cdot\lambda$ with $w_i\in W_{[\lambda]}$. By exercise 1.1 (b), all weights of $M_b^i$ lie in $[w_i\cdot\lambda]$ for each $i$. Since $w_i\in W_{[\lambda]}$, we get $w_i\lambda-w_j\lambda\in\Lambda_r$ and then $[w_i\cdot\lambda]=[w_j\cdot\lambda]$ for any $i,j$. Hence all weight of $M_b$ lie in $[w\cdot\lambda]$ where $w\in W_{[\lambda]}$.

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