"Talk on BGG resolution"의 두 판 사이의 차이
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| (다른 사용자 한 명의 중간 판 166개는 보이지 않습니다) | |||
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==characters== | ==characters== | ||
| − | * let | + | * let <math>\lambda\in \mathfrak{h}^*</math> |
| − | + | :<math> | |
\operatorname{ch} M({\lambda})=\frac{e^{\lambda}}{\prod_{\alpha>0}(1-e^{-\alpha})} | \operatorname{ch} M({\lambda})=\frac{e^{\lambda}}{\prod_{\alpha>0}(1-e^{-\alpha})} | ||
| − | + | </math> | |
| − | * let | + | * let <math>\lambda\in \Lambda^+</math> |
;thm ([[Weyl-Kac character formula|Weyl character formula]]) | ;thm ([[Weyl-Kac character formula|Weyl character formula]]) | ||
:<math> | :<math> | ||
| 10번째 줄: | 10번째 줄: | ||
</math> | </math> | ||
* thus we have | * thus we have | ||
| − | + | :<math> | |
\operatorname{ch}L(\lambda)=\sum_{w \in W}(-1)^{\ell(w)}\operatorname{ch} M(w\cdot \lambda) \label{WCF} | \operatorname{ch}L(\lambda)=\sum_{w \in W}(-1)^{\ell(w)}\operatorname{ch} M(w\cdot \lambda) \label{WCF} | ||
| − | + | </math> | |
;prop | ;prop | ||
| − | If | + | If <math>0\to M' \to M \to M'' \to 0</math> is a short exact sequence in <math>\mathcal{O}</math>, we have |
| − | + | :<math> | |
\operatorname{ch}M=\operatorname{ch}M'+\operatorname{ch}M'' | \operatorname{ch}M=\operatorname{ch}M'+\operatorname{ch}M'' | ||
| − | + | </math> | |
or | or | ||
| − | + | :<math> | |
\operatorname{ch}M'-\operatorname{ch}M+\operatorname{ch}M''=0 | \operatorname{ch}M'-\operatorname{ch}M+\operatorname{ch}M''=0 | ||
| − | + | </math> | |
* if we have a long exact sequence, we still get a similar alternating sum = 0 | * if we have a long exact sequence, we still get a similar alternating sum = 0 | ||
** why? [[Euler-Poincare principle|Euler-Poincare mapping]] : a long exact sequence can be decomposed into short exact sequences. | ** why? [[Euler-Poincare principle|Euler-Poincare mapping]] : a long exact sequence can be decomposed into short exact sequences. | ||
| − | ** then the Euler characteristic of a resolution makes sense | + | ** then the Euler characteristic of a finite resolution makes sense |
| − | * goal : realize the alternating sum \ref{WCF} as an Euler characteristic of a suitable resolution of | + | * goal : realize the alternating sum \ref{WCF} as an Euler characteristic of a suitable resolution of <math>L(\lambda)</math> |
| − | * The BGG resolution resolves a finite-dimensional simple | + | * The BGG resolution resolves a finite-dimensional simple <math>\mathfrak{g}</math>-module <math>L(\lambda)</math> by direct sums of Verma modules indexed by weights "of the same length" in the orbit <math>W\cdot \lambda</math> |
;thm (Bernstein-Gelfand-Gelfand Resolution) | ;thm (Bernstein-Gelfand-Gelfand Resolution) | ||
| − | Fix | + | Fix <math>\lambda\in \Lambda^{+}</math>. There is an exact sequence of Verma modules |
| − | + | :<math> | |
0 \to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots \to M({\lambda})\to L({\lambda})\to 0 | 0 \to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots \to M({\lambda})\to L({\lambda})\to 0 | ||
| − | + | </math> | |
| − | where | + | where <math>\ell(w)</math> is the length of the Weyl group element <math>w</math>, <math>w_0</math> is the Weyl group element |
| − | of maximal length. Here | + | of maximal length. Here <math>\rho</math> is half the sum of the positive roots. |
==example of BGG resolution== | ==example of BGG resolution== | ||
| − | === | + | ===<math>\mathfrak{sl}_2</math>=== |
* <math>L({\lambda})</math> : irreducible highest weight module | * <math>L({\lambda})</math> : irreducible highest weight module | ||
** weights <math>\lambda ,-2+\lambda ,\cdots, -\lambda</math> | ** weights <math>\lambda ,-2+\lambda ,\cdots, -\lambda</math> | ||
| 43번째 줄: | 43번째 줄: | ||
** weights <math>\lambda ,-2+\lambda ,\cdots, -\lambda, -\lambda-2,\cdots</math> | ** weights <math>\lambda ,-2+\lambda ,\cdots, -\lambda, -\lambda-2,\cdots</math> | ||
;thm | ;thm | ||
| − | If | + | If <math>\lambda\in \Lambda^+</math>, the maximal submodule <math>N(\lambda)</math> of <math>M(\lambda)</math> is the sum of submodules <math>M(s_i\cdot \lambda)</math> for <math>1\le i \le l</math>, where <math>l</math> is the rank of <math>\mathfrak{g}</math>. |
| − | * | + | * <math>s_{1}(\lambda+\rho)=-\lambda-\rho</math>, <math>s_{1}\cdot \lambda=-\lambda-2\rho</math> |
| − | * if we identity | + | * if we identity <math>\Lambda = \mathbb{Z} \omega_1</math> with <math>\mathbb{Z}</math>, then <math>\rho=1,\alpha=2</math> |
* we have | * we have | ||
| − | + | :<math>L({\lambda})=M({\lambda})/M({-\lambda-2})</math> or | |
:<math>0\to M({-\lambda-2})\to M({\lambda})\to L({\lambda})\to 0</math> | :<math>0\to M({-\lambda-2})\to M({\lambda})\to L({\lambda})\to 0</math> | ||
* this gives a BGG resolution | * this gives a BGG resolution | ||
| − | * character of | + | * character of <math>L({\lambda})</math> = alternating sum of characters of Verma modules |
:<math>\operatorname{ch}{L({\lambda})}=\operatorname{ch}{M({\lambda})}-\operatorname{ch}{M({-\lambda-2})}=\frac{e^{\lambda}}{1-e^{-2}}-\frac{e^{-\lambda-2}}{1-e^{-2}}</math> | :<math>\operatorname{ch}{L({\lambda})}=\operatorname{ch}{M({\lambda})}-\operatorname{ch}{M({-\lambda-2})}=\frac{e^{\lambda}}{1-e^{-2}}-\frac{e^{-\lambda-2}}{1-e^{-2}}</math> | ||
* comparison with [[Weyl-Kac character formula]] | * comparison with [[Weyl-Kac character formula]] | ||
| 59번째 줄: | 59번째 줄: | ||
==weak BGG resolution== | ==weak BGG resolution== | ||
;def | ;def | ||
| − | * We say that | + | * We say that <math>M \in O</math> has a standard filtration (also called a Verma flag) if there is a sequence of submodules |
| − | + | :<math>0 = M_0 \subset M_1 \subset M_2 \subset \cdots \subset M_n = M</math> | |
| − | for which each | + | for which each <math>M^i := M_i/M_{i−1}\, (1 \le i \le n)</math> is isomorphic to a Verma module. |
;thm (Weak BGG resolution) | ;thm (Weak BGG resolution) | ||
There is an exact sequence | There is an exact sequence | ||
| − | + | :<math> | |
0 \to M({w_0\cdot \lambda}) = D_m^{\lambda} \to D_{m-1}^{\lambda} \to \cdots \to D_1^{\lambda} \to D_0^{\lambda}=M(\lambda) \to L(\lambda) \to 0 | 0 \to M({w_0\cdot \lambda}) = D_m^{\lambda} \to D_{m-1}^{\lambda} \to \cdots \to D_1^{\lambda} \to D_0^{\lambda}=M(\lambda) \to L(\lambda) \to 0 | ||
| − | + | </math> | |
| − | where | + | where <math>D_{k}^{\lambda}</math> has a standard filtration involving exactly once each of the Verma modules <math>M(w\cdot \lambda)</math> with <math>\ell(w)=k</math> |
===strategy to construct a BGG resolution=== | ===strategy to construct a BGG resolution=== | ||
| − | + | # construct a relative version of standard resoultion <math>D_k:=U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\Lambda^{k}(\mathfrak{g}/\mathfrak{b})</math> for <math>L(0)</math> | |
| − | + | # construct a weak BGG resolution <math>D_k^0:=D_k^{\chi_{0}}</math> for <math>L(0)</math> by cutting down to the principal block component of each term | |
| − | + | # construct a weak BGG resolution <math>D_k^\lambda : = (D_k^0\otimes L(\lambda))^{\chi_{\lambda}}</math> for <math>L(\lambda)</math> (we can also do this by applying the translation functor) | |
| − | + | # show that it is actually a BGG resolution by computing <math>\operatorname{Ext}</math> between Verma modules | |
==standard resolution of trivial module== | ==standard resolution of trivial module== | ||
| − | * free | + | * free <math>U(\mathfrak{g})</math>-modules <math>U(\mathfrak{g})\otimes_{\mathbb{C}}\wedge^{k}(\mathfrak{g})</math> |
* standard resolution of trivial module in [[Lie algebra cohomology]] | * standard resolution of trivial module in [[Lie algebra cohomology]] | ||
| − | + | :<math>\cdots \to U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{k}(\mathfrak{g})\to U(\mathfrak{g})\otimes_{\mathbb{C}}\wedge^{k-1}(\mathfrak{g})\to \cdots \to U(\mathfrak{g})\otimes_{\mathbb{C}}\wedge^{0}(\mathfrak{g})\to L(0)</math> | |
| − | * the sequence of modules | + | * the sequence of modules <math>D_k:=U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\wedge^{k}(\mathfrak{g}/\mathfrak{b})</math> is a relative version of the standard resolution |
| − | * we can describe | + | * we can describe <math>D_0</math> and <math>D_m</math> explicitly |
| − | * define | + | * define <math>U(\mathfrak{g})</math>-module homomorphism <math>\partial_k : D_k \to D_{k-1}</math> as |
| − | + | :<math> | |
\begin{align} | \begin{align} | ||
| − | \partial_k ( u\otimes \xi_1 \wedge \cdots \xi _k): &= \sum_{i=1}^k(-1)^{i+1}(uz_i\otimes \xi_1\wedge \cdots \hat{\xi_i}\wedge \cdots \xi_k)\\ | + | \partial_k ( u\otimes \xi_1 \wedge \cdots \wedge \xi _k): &= \sum_{i=1}^k(-1)^{i+1}(uz_i\otimes \xi_1\wedge \cdots \wedge \hat{\xi_i}\wedge \cdots \wedge\xi_k)\\ |
| − | &+\sum_{1\le i<j \le k} (-1)^{i+j}(u \otimes \overline{[z_iz_j]}\wedge \xi_1\wedge \cdots \hat{\xi_i}\wedge \cdots \hat{\xi_j} \cdots \xi_k) | + | &+\sum_{1\le i<j \le k} (-1)^{i+j}(u \otimes \overline{[z_iz_j]}\wedge \xi_1\wedge \cdots \wedge \hat{\xi_i}\wedge \cdots \wedge\hat{\xi_j} \wedge \cdots \xi_k) |
\end{align} | \end{align} | ||
| − | + | </math> | |
| − | * need to show that it is actually a complex and exact | + | where <math>z_i\in \mathfrak{g}</math> is a representative of <math>\xi_i\in \mathfrak{g}/\mathfrak{b}</math> and <math>\overline{z}</math> denotes the canonical surjection <math>z\in\mathfrak{g}</math> into <math>\mathfrak{g}/\mathfrak{b}</math>. |
| + | * need to show that <math>\partial_k</math> is well-defined and it is actually a complex and exact | ||
| + | ===exactness=== | ||
| + | * exactness is tricky | ||
| + | ** see Wallach, Real Reductive Groups I 6.A | ||
** see Knapp, Lie Groups, Lie Algebras, and Cohomology IV.6 | ** see Knapp, Lie Groups, Lie Algebras, and Cohomology IV.6 | ||
| + | * Let <math>U_j(\mathfrak{g}):=U^j(\mathfrak{g})U(\mathfrak{b})</math> where <math>U^j(\mathfrak{g})</math> is the span of the PBW basis whose degree is <math>\le j</math> | ||
| + | * note that <math>U_j(\mathfrak{g})=S_j(\mathfrak{n}^-)U(\mathfrak{b})</math> where <math>S_j(\mathfrak{n}^-)=\sum_{0\le k\le j}S^k(\mathfrak{n}^-)</math>. | ||
| + | * <math>S^k(\mathfrak{n}^-)</math> denote the elements of <math>\operatorname{Sym}(\mathfrak{n}^-)</math> that are homogeneous of degree <math>k</math>. | ||
| + | * <math>E_{j,k}:=U_j(\mathfrak{g})\otimes_{U(\mathfrak{b})}\wedge^{k}(\mathfrak{g}/\mathfrak{b})</math> | ||
| + | * <math>\{E_{j,k}\}_{j\ge 0}</math> gives a filtration of <math>D_k</math> | ||
| + | * let <math>\partial_0 : D_0\to L(0)</math> be the canonical surjection | ||
| + | * '''exercise''' : <math>\partial_k : E_{j,k}\to E_{j+1,k-1}, k\ge 1</math> | ||
| + | * <math>\partial_k, k\ge 1</math> induces <math>\overline{\partial_k} : E_{j,k}/E_{j-1,k}\to E_{j+1,k-1}/E_{j,k-1}</math> | ||
| + | |||
| + | ;prop | ||
| + | For each <math>j\ge 1</math>, <math>\{(E_{j,k}/E_{j-1,k},\overline{\partial_k})\}_{-1\le k\le m+1}</math> is an exact sequence. (here <math>-1</math> and <math>m+1</math> terms are zero) | ||
| − | |||
| − | |||
| − | |||
| − | |||
;proof | ;proof | ||
| − | + | As a vector space, <math>E_{j,k}/E_{j-1,k}\cong S^j(\mathfrak{n}^-)\otimes \wedge^{k}(\mathfrak{n}^-)</math>. | |
| + | |||
| + | Now we can apply basic results on the Koszul complexes : | ||
| − | ; | + | ;theorem |
| − | + | For each <math>j\geq 1</math>, the following is exact | |
| + | :<math> | ||
| + | 0\to S^{j-m}(V)\otimes \wedge^m(V) \to S^{j-m+1}(V)\otimes \wedge^{m-1}(V) \to \cdots \to S^{j-1}(V)\otimes \wedge^{1}(V) \to S^{j}(V)\to 0 | ||
| + | </math> | ||
| + | where <math>S^j(V)=0</math> for <math>j<0</math> | ||
| + | * see Lang, Algebra ' Koszul complex' | ||
| + | |||
| + | * examples | ||
| + | :<math> | ||
| + | 0\to \wedge^2 \to S^1\otimes \wedge^1 \to S^2\to 0 | ||
| + | </math> | ||
| + | or | ||
| + | :<math> | ||
| + | 0\to E_{0,2} \to E_{1,1}/E_{0,1} \to E_{2,0}/E_{1,0} \to 0 | ||
| + | </math> | ||
| + | |||
| + | :<math> | ||
| + | 0\to \wedge^1 \to S^1 \to 0 | ||
| + | </math> | ||
| + | or | ||
| + | :<math> | ||
| + | 0\to E_{0,1} \to E_{1,0}/E_{0,0} \to 0 | ||
| + | </math> | ||
| + | * in fact, we also have | ||
| + | :<math> | ||
| + | 0\to E_{0,0}\to L(0)\to 0 | ||
| + | </math> | ||
| + | ;prop | ||
| + | For each <math>j\ge 1</math>, <math>\{(E_{j,k},\partial_k)\}_{-1\le k \le m+1}</math> is exact. (here <math>-1</math> and <math>m+1</math> terms are zero) | ||
;proof | ;proof | ||
| − | + | Suppose <math>u\in E_{j,k},\, j,k\geq 1</math> satisfies <math>\partial(u)=0</math> in <math>E_{j+1,k-1}</math>. | |
| + | |||
| + | show : there exists <math>v\in E_{j-1,k+1}</math> such that <math>\partial(v)=u</math> in | ||
| + | :<math> | ||
| + | E_{j-1,k+1} \to E_{j,k}\to E_{j+1,k-1}. | ||
| + | </math> | ||
| − | + | We look at | |
| + | :<math> | ||
| + | E_{j-1,k+1}/E_{j-2,k+1} \to E_{j,k}/E_{j-1,k} \to E_{j+1,k-1}/E_{j,k}. | ||
| + | </math> | ||
| + | As <math>\overline{\partial}(\overline{u})=0</math>, we can find <math>v_1\in E_{j-1,k+1}</math> such that <math>\overline{\partial}(\overline{v_1})=\overline{u}</math>. | ||
| − | + | As <math>\overline{u}-\overline{\partial}(\overline{v_1})=0</math> in <math>E_{j,k,}/E_{j-1,k}</math>, there exists <math>w\in E_{j-1,k}</math> such that <math>u-\partial(v_1)=w</math>. | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | \ | ||
| − | |||
| − | |||
| − | |||
| − | == | + | Now <math>\partial(w)=0</math> in <math>E_{j,k-1}</math> and by the same argument applied to |
| + | :<math> | ||
| + | E_{j-2,k+1} \to E_{j-1,k}\to E_{j,k-1}, | ||
| + | </math> | ||
| + | there exists <math>v_2\in E_{j-2,k+1}</math> such that <math>w-\partial (v_2)\in E_{j-2,k}</math>, i.e. <math>u-\partial(v_1)-\partial(v_2)\ \in E_{j-2,k}</math>. | ||
| + | |||
| + | By repeating this, we can find <math>v_1,\cdots, v_j</math> such that each <math>v_i\in E_{j-i,k}</math> and <math>u-\partial(v_1)-\cdots -\partial(v_j)\ \in E_{0,k}</math>. | ||
| + | |||
| + | As <math>\overline{\partial} : E_{0,k}\to E_{1,k-1}/E_{0,k-1}</math> is injective and <math>\partial(u-\partial(v_1)-\cdots -\partial(v_j))=0 \in E_{1,k-1}</math>, we can conclude <math>u-\partial(v_1)-\cdots -\partial(v_j)=0</math> or, | ||
| + | :<math> | ||
| + | u=\partial(v_1)+\cdots +\partial(v_j). | ||
| + | </math> | ||
| + | Thus if we set <math>v:=v_1+\cdots +v_j\in E_{j-1,k+1}</math> and <math>\partial(v)=u</math>. ■ | ||
| + | |||
| + | |||
| + | ;thm | ||
| + | <math>0\to D_m \to \cdots \to D_k\to \cdots \to D_1 \to D_0 \to L(0)\to 0</math> is exact. | ||
| + | ;proof | ||
| + | The above proposition clearly proves the exactness at <math>D_k,\, k\ge 1</math>. | ||
| + | |||
| + | Assume that <math>u\in D_0</math>, hence <math>u\in E_{j,0}</math> for some <math>j</math>. Let <math>u=u_0+u_1</math> where <math>u_0</math> is the degree zero piece and <math>u_1</math> other terms in PBW basis. | ||
| + | |||
| + | If <math>\partial(u)=0</math>, then it implies <math>u_0 = 0</math> as only degree zero term survives under <math>\partial</math>. | ||
| + | |||
| + | Therefore <math>j\ge 1</math> and the above proposition still applies to conclude that there exists <math>v\in E_{j-1,1}</math> such that <math>\partial(v)=u</math>. | ||
| + | |||
| + | This proves the exactness at <math>D_0</math>. ■ | ||
| + | |||
| + | ==weak BGG resolution of <math>L(0)</math>== | ||
| + | * goal : find standard filtrations of <math>D_k</math> and <math>D_k^0</math> and their Verma subquotients | ||
| + | ;lemma | ||
| + | Let <math>N</math> be a finite-dimensional <math>U(\mathfrak{b})</math>-module having a basis of weight vectors. Then <math>M=U(\mathfrak{g})\otimes_{U(\mathfrak{b})} N</math> has a standard filtration and each weight of <math>N</math> gives a corresponding Verma subquotient. | ||
| + | ;proof | ||
| + | Let <math>\{v_1,\cdots, v_r \}</math> be a basis of <math>N</math> consisting of weight vectors and let <math>\mu_i</math> be the weight of <math>v_i</math>. | ||
| + | |||
| + | We order the basis so that <math>i\le j</math> whenever <math>\mu_i\le \mu_j</math> | ||
| + | |||
| + | Let <math>N_k</math> be a space spanned by <math>\{v_k,\cdots, v_r \}</math> for <math>1\le k \le r</math>. | ||
| + | |||
| + | '''exercise'''. Check that each <math>N_k</math> is a <math>U(\mathfrak{b})</math>-submodule. (hint : weight cannot decrease under <math>U(\mathfrak{b})</math> action) | ||
| + | |||
| + | We have a flag of <math>U(\mathfrak{b})</math>-modules : | ||
| + | :<math> | ||
| + | 0 \subset N_r \subset N_{r-1} \subset \cdots \subset N_1 = N \label{Nflag} | ||
| + | </math> | ||
| + | |||
| + | We get a standard filtration of <math>M</math> from \ref{Nflag} as the functor <math>N\mapsto U(\mathfrak{g})\otimes_{U(\mathfrak{b})} N</math> is exact. (See Remark 1.3) ■ | ||
| + | * this lemma is true even if we drop the assumption about the existence of basis of weight vectors | ||
| + | * but such module induces a module not necessarily on the BGG category | ||
| + | ;prop | ||
| + | <math>D_k</math> has a standard filtration with Verma subquotients associated to sums of <math>k</math> distinct negative roots. | ||
| + | ;proof | ||
| + | If we apply the above lemma, enough to answer : | ||
| + | |||
| + | Q. what are the weights of <math>\wedge^k (\mathfrak{g}/\mathfrak{b})</math> as <math>\mathfrak{b}</math>-module? | ||
| + | |||
| + | A : sum of <math>k</math> distinct negative roots | ||
| + | ■ | ||
| + | |||
| + | ;prop | ||
| + | <math>D_k^0</math> has a standard filtration with Verma subquotients <math>M(w\cdot 0), w\in W^{(k)}</math> where <math>W^{(k)}:=\{w\in W|\ell(w)=k\}</math> | ||
| + | |||
| + | ;proof | ||
| + | Taking a block preserves exactness : | ||
| + | |||
| + | If <math>0\to M' \to M \to M(\mu)\to 0</math>, then <math>0\to (M')^0 \to M^0 \to (M(\mu))^0 \to 0</math>. | ||
| + | |||
| + | :<math> | ||
| + | (M(\mu))^0 = | ||
| + | \begin{cases} | ||
| + | M(\mu) , & \text{if </math>\mu<math> is linked to </math>0<math> (</math>\mu=w\cdot 0<math> for some </math>w\in W<math>)}\\ | ||
| + | 0, & \text{otherwise} \\ | ||
| + | \end{cases} | ||
| + | </math> | ||
| + | |||
| + | Thus we obtain a standard filtration of <math>D_k^0</math> from that of <math>D_k</math>. | ||
| + | |||
| + | What are the Verma subquotients or when is <math>\beta</math> linked to <math>0</math> if <math>\beta</math> is given a sum of <math>k</math> distinct negative roots? | ||
| + | |||
| + | '''fact''' : <math>\ell(w)=|w \Phi^+ \cap \Phi^-|</math>. | ||
| + | |||
| + | '''exercise''' : Let <math>\beta_w:=w\cdot 0</math> for each <math>w\in W</math>. Then <math>\beta_w</math> is a sum of elements in <math>w \Phi^+ \cap \Phi^-</math>. | ||
| + | |||
| + | '''exercise''' : Let <math>\Pi \subset \Phi^-</math> be given. If the sum <math>\beta</math> of elements of <math>\Pi</math> is <math>\beta_w</math> for some <math>w\in W</math>, then <math>\Pi = w \Phi^+ \cap \Phi^-</math>. (we know the whole set by only looking at the sum of them) | ||
| + | |||
| + | Thus <math>\beta</math> is a sum of <math>k</math> distinct negative roots and linked to <math>0</math> iff there exists <math>w\in W^{(k)}</math>. | ||
| + | |||
| + | Finally, | ||
| + | |||
| + | '''fact''' : <math>|W\cdot 0|=|W|</math>. (this implies each <math>\beta_w</math> is distinct) | ||
| + | |||
| + | Therefore each <math>M(w\cdot 0),\, w\in W^{(k)}</math> appears only in once our standard filtration. | ||
| + | |||
| + | ■ | ||
| + | * thus we have found a weak BGG resolution of <math>L(0)</math> | ||
| + | |||
| + | ==weak BGG resolution of <math>L(\lambda)</math>== | ||
* based on Remark in 6.2 | * based on Remark in 6.2 | ||
| − | * | + | * let <math>\lambda\in \Lambda^+</math> |
| − | + | * goal : find a standard filtration of <math>D_k^\lambda : = (D_k^0\otimes L(\lambda))^{\chi_{\lambda}}</math> and its Verma subquotients | |
| − | ; | + | ;prop |
| − | + | <math>D_k^0\otimes L(\lambda)</math> has a standard filtration with Verma subquotients <math>M(w\cdot 0 + \mu)</math> where <math>w\in W^{(k)}</math> and <math>\mu</math> is a weight of <math>L(\lambda)</math>. | |
;proof | ;proof | ||
| − | Use | + | Use the following : |
;thm (3.6) | ;thm (3.6) | ||
| − | Let | + | Let <math>M</math> be a finite dimensional <math>U(\mathfrak{g})</math>-module. For any <math>\lambda\in \mathfrak{h}^{*}</math>, <math>T:=M(\lambda)\otimes M</math> has a standard filtration with Verma subquotients <math>M(\lambda+\mu)</math>. Here <math>\mu</math> ranges over the weights of <math>M</math>, each occurring <math>\dim M_{\mu}</math> times in the filtration. |
| + | |||
| + | Tensoring with a finite-dimensional representation is an exact functor in BGG category (thm 1.1). | ||
| + | |||
| + | If <math>0\to N\to M \to M(\lambda)\to 0</math>, then <math>0\to N\otimes L(\lambda)\to M\otimes L(\lambda) \to M(\lambda)\otimes L(\lambda)\to 0</math> | ||
| + | Use this to construct a standard filtration on <math>D_k^0 \otimes L(\lambda)</math> from that of <math>D_k^0</math>. ■ | ||
| − | ; | + | ;prop |
| − | + | <math>D_k^\lambda</math> has a standard filtration with Verma subquotients <math>M(w\cdot \lambda),\, w\in W^{k}</math>. | |
;proof | ;proof | ||
| − | + | Again taking the block component for <math>\chi_{\lambda}</math> is exact and it gives a standard filtration of <math>D_k^\lambda</math> from that of <math>D_k^0\otimes L(\lambda)</math>. | |
| + | |||
| + | We need to determine when <math>w\cdot 0 + \mu</math> is linked to <math>\lambda</math>. | ||
| − | + | '''exercise''' : <math>w\cdot 0 + \mu</math> is linked to <math>\lambda</math> iff <math>\mu = w\lambda</math>. | |
| − | + | As <math>\lambda</math> is the highest weight in <math>L(\lambda)</math>, each <math>w\lambda</math> is with weight multiplicity 1. Thus each <math>M(w\cdot \lambda),\, w\in W^{(k)}</math> appears only once in our standard filtration. Note that each <math>w\cdot \lambda</math> is distinct as : | |
| − | + | '''fact''' : <math>|W\cdot \lambda|=|W|</math> for <math>\lambda\in \Lambda^+</math>.■ | |
| − | |||
| − | |||
==extensions of Verma modules== | ==extensions of Verma modules== | ||
| − | * | + | * we have constructed a weak BGG resolution of <math>L(\lambda)</math> involving <math>D_k^{\lambda}</math> |
| − | + | * goal : <math>D_k^{\lambda}</math> is a direct sum of Verma modules | |
| − | + | ||
| − | ;def ( | + | ;def |
| − | Define a partial | + | Let <math>\mu, \lambda\in \mathfrak{h}^{*}</math>. Write <math>\mu \uparrow \lambda</math> if <math>\mu = \lambda</math> or there exists <math>\alpha\in \Phi^+</math> such that <math>\mu=s_{\alpha}\cdot \lambda < \lambda </math> (<math>\mathbb{Z}^+</math>-linear combination of simple roots) |
| + | |||
| + | We say <math>\mu</math> is ''strongly linked'' to <math>\lambda</math> if <math>\mu = \lambda</math> or there exist <math>\alpha_1,\cdots, \alpha_r\in \Phi^+</math> such that <math>\mu=(s_{\alpha_1}\cdots s_{\alpha_r})\cdot \lambda \uparrow (s_{\alpha_2}\cdots s_{\alpha_r})\cdot \lambda \uparrow \cdots \uparrow ( s_{\alpha_r})\cdot \lambda \uparrow \lambda </math> | ||
| + | |||
| + | ;def | ||
| + | Let <math>w,w'\in W</math>. Write <math>w'\xrightarrow{t} w</math> whenever <math>w = t w' </math> for some reflection <math>t\in W</math> and <math>\ell(w') < \ell(w)</math>. Define <math>w'<w</math> if there is a sequence <math>w'=w_0\to w_1\to \cdots \to w_n=w</math>. Extend this relation to a partial ordering of <math>W</math> and call it the ''[[Bruhat ordering]]''. | ||
| + | |||
| + | example : http://groupprops.subwiki.org/wiki/File:Bruhatons3.png draw the edge of only the difference of length is 1. | ||
| − | The | + | '''exercise''' : Let <math>w\in W, \alpha \in \Phi^+</math> be given. The following are equivalent : |
| − | + | ||
| + | (i) There exists a <math>\lambda\in \Lambda^{+}</math> such that <math>s_{\alpha}\cdot (w\cdot \lambda) \uparrow w\cdot \lambda</math> | ||
| + | |||
| + | (ii) <math>s_{\alpha}w > w</math> | ||
| + | |||
| + | hint : use | ||
| + | |||
| + | '''fact''' : <math>w^{-1}\alpha>0</math> iff <math>\ell(s_{\alpha}w)> \ell(w)</math>. | ||
;thm | ;thm | ||
| − | Let | + | (a) Let <math>\lambda,\mu\in \mathfrak{h}^{*}</math>. If <math>\operatorname{Ext}_{\mathcal{O}}(M(\mu),M(\lambda))\neq 0</math>, then <math>\mu</math> is strongly linked to <math>\lambda</math> but <math>\mu \neq \lambda</math> |
| − | ( | + | (b) Let <math>\lambda\in \Lambda^{+}</math> and <math>w,w'\in W</math>. If <math>\operatorname{Ext}_{\mathcal{O}}(M(w'\cdot\lambda),M(w\cdot\lambda))\neq 0</math>, then <math>w<w'</math> in the [[Bruhat ordering]]. In particular, <math>\ell(w)<\ell(w')</math>. |
| − | (b) | + | ;proof of (a) |
| − | + | (a) uses projective cover, BGG reciprocity and BGG theorem from the previous chapters. (so we skip the proof) ■ | |
| − | + | ||
| + | ;proof of (b) | ||
| + | From (a), we see that <math>w'\cdot\lambda</math> is strongly linked to <math>w\cdot\lambda</math>. Thus we can find | ||
| + | <math> | ||
| + | w'\cdot\lambda=(s_{\alpha_1}\cdots s_{\alpha_r})\cdot (w\cdot\lambda) \uparrow (s_{\alpha_2}\cdots s_{\alpha_r})\cdot (w\cdot\lambda) \uparrow \cdots \uparrow ( s_{\alpha_r})\cdot (w\cdot\lambda) \uparrow (w\cdot\lambda) | ||
| + | </math> | ||
| + | From exercise, <math>w'=s_{\alpha_1}\cdots s_{\alpha_r}w> \cdots > s_{\alpha_r}w > w</math>. | ||
| + | ■ | ||
| + | |||
| + | ==finish== | ||
| + | ;prop | ||
| + | A weak BGG resolution is a BGG resolution. | ||
| + | ;proof | ||
| + | |||
| + | Use induction on the length of standard filtration. | ||
| + | |||
| + | Let <math>M\subset D_k^{\lambda}</math> and <math>D_k^{\lambda}/M \cong M(w'\cdot \lambda)</math> for some <math>w'\in W^{(k)}</math>. | ||
| + | |||
| + | By induction hypothesis, <math>0\to M=\oplus_{w\in W^{(k)},w\neq w'} M(w\cdot \lambda)\to D_k^{\lambda} \to M(w'\cdot \lambda) \to 0</math>, then this splits as <math>\operatorname{Ext}\left(M(w'\cdot \lambda ),\oplus M(w\cdot \lambda)\right)</math> is zero (ext is additive). | ||
| + | |||
| + | ■ | ||
==memo== | ==memo== | ||
* proof of Thm 3.6 uses the following (we don't need this for our goal) | * proof of Thm 3.6 uses the following (we don't need this for our goal) | ||
;thm Tensor Identity (56p) | ;thm Tensor Identity (56p) | ||
| − | Let | + | Let <math>M</math> be a <math>U(\mathfrak{g})</math>-module and <math>L</math> a <math>U(\mathfrak{b})</math>-module. Then |
| − | + | :<math> | |
(U(\mathfrak{g})\otimes_{U(\mathfrak{b})}L)\otimes M \cong U(\mathfrak{g})\otimes_{U(\mathfrak{b})}(L \otimes M) | (U(\mathfrak{g})\otimes_{U(\mathfrak{b})}L)\otimes M \cong U(\mathfrak{g})\otimes_{U(\mathfrak{b})}(L \otimes M) | ||
| − | + | </math> | |
;prop (3.7) | ;prop (3.7) | ||
| − | Let | + | Let <math>M\in \mathcal{O}</math> have a standard filtration. If <math>\lambda</math> is maximal among the weights of <math>M</math>, then <math>M</math> has a submodule isomorphic to <math>M(\lambda)</math> and <math>M/M(\lambda)</math> has a standard filtration. |
===overview=== | ===overview=== | ||
* principal block : filtering through central characters | * principal block : filtering through central characters | ||
| − | ** is a block a | + | ** is a block a <math>U(\mathfrak{g})</math>-submodule? yes |
** how to check that it preserves the exactness : any homomorphism between modules belonging to different blocks will be zero | ** how to check that it preserves the exactness : any homomorphism between modules belonging to different blocks will be zero | ||
| − | ** how to describe | + | ** how to describe <math>\chi_{\lambda}</math>? use the twisted Harish-Chandra homomorphism <math>\psi : Z(\mathfrak{g})\to S(\mathfrak{h})</math>. we have |
| − | + | :<math> | |
\chi_{\lambda}(z)=(\lambda+\rho)(\psi(z)),\quad z\in Z(\mathfrak{g}) | \chi_{\lambda}(z)=(\lambda+\rho)(\psi(z)),\quad z\in Z(\mathfrak{g}) | ||
| − | + | </math> | |
| − | ** see 26p for an example of | + | ** see 26p for an example of <math>\chi_{\lambda}</math> in type <math>A_1</math> |
* combinatorial results | * combinatorial results | ||
| − | ** longest elements satisfies | + | ** longest elements satisfies <math>w_0\cdot 0 = -2\rho</math> (related to diagram automorphism) |
| − | ** consider the set of sum of k distinct roots. Which elements are linked to | + | ** consider the set of sum of k distinct roots. Which elements are linked to <math>0</math>? |
** Bruhat ordering | ** Bruhat ordering | ||
* Bruhat ordering and strong linkage relation | * Bruhat ordering and strong linkage relation | ||
| − | ** let | + | ** let <math>\lambda \in \Lambda^+</math> (which is regular for the dot-action of <math>W</math>) |
| − | ** | + | ** <math>w'\cdot \lambda< w \cdot \lambda </math> translates into <math>w < w'</math> in the Bruhat ordering |
* strong linkage relation and extension of Verma modules | * strong linkage relation and extension of Verma modules | ||
* for exterior powers, see [[Lie Algebras of Finite and Affine Type by Carter]] | * for exterior powers, see [[Lie Algebras of Finite and Affine Type by Carter]] | ||
| 200번째 줄: | 377번째 줄: | ||
[[분류:Talks and lecture notes]] | [[분류:Talks and lecture notes]] | ||
[[분류:abstract concepts]] | [[분류:abstract concepts]] | ||
| + | [[분류:migrate]] | ||
2020년 11월 14일 (토) 02:14 기준 최신판
characters
- let \(\lambda\in \mathfrak{h}^*\)
\[ \operatorname{ch} M({\lambda})=\frac{e^{\lambda}}{\prod_{\alpha>0}(1-e^{-\alpha})} \]
- let \(\lambda\in \Lambda^+\)
- thm (Weyl character formula)
\[ \operatorname{ch}L({\lambda})=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot \lambda}}{\prod_{\alpha>0}(1-e^{-\alpha})} \]
- thus we have
\[ \operatorname{ch}L(\lambda)=\sum_{w \in W}(-1)^{\ell(w)}\operatorname{ch} M(w\cdot \lambda) \label{WCF} \]
- prop
If \(0\to M' \to M \to M'' \to 0\) is a short exact sequence in \(\mathcal{O}\), we have \[ \operatorname{ch}M=\operatorname{ch}M'+\operatorname{ch}M'' \] or \[ \operatorname{ch}M'-\operatorname{ch}M+\operatorname{ch}M''=0 \]
- if we have a long exact sequence, we still get a similar alternating sum = 0
- why? Euler-Poincare mapping : a long exact sequence can be decomposed into short exact sequences.
- then the Euler characteristic of a finite resolution makes sense
- goal : realize the alternating sum \ref{WCF} as an Euler characteristic of a suitable resolution of \(L(\lambda)\)
- The BGG resolution resolves a finite-dimensional simple \(\mathfrak{g}\)-module \(L(\lambda)\) by direct sums of Verma modules indexed by weights "of the same length" in the orbit \(W\cdot \lambda\)
- thm (Bernstein-Gelfand-Gelfand Resolution)
Fix \(\lambda\in \Lambda^{+}\). There is an exact sequence of Verma modules \[ 0 \to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots \to M({\lambda})\to L({\lambda})\to 0 \] where \(\ell(w)\) is the length of the Weyl group element \(w\), \(w_0\) is the Weyl group element of maximal length. Here \(\rho\) is half the sum of the positive roots.
example of BGG resolution
\(\mathfrak{sl}_2\)
- \(L({\lambda})\) : irreducible highest weight module
- weights \(\lambda ,-2+\lambda ,\cdots, -\lambda\)
- \(M({\lambda})\) : Verma modules
- weights \(\lambda ,-2+\lambda ,\cdots, -\lambda, -\lambda-2,\cdots\)
- thm
If \(\lambda\in \Lambda^+\), the maximal submodule \(N(\lambda)\) of \(M(\lambda)\) is the sum of submodules \(M(s_i\cdot \lambda)\) for \(1\le i \le l\), where \(l\) is the rank of \(\mathfrak{g}\).
- \(s_{1}(\lambda+\rho)=-\lambda-\rho\), \(s_{1}\cdot \lambda=-\lambda-2\rho\)
- if we identity \(\Lambda = \mathbb{Z} \omega_1\) with \(\mathbb{Z}\), then \(\rho=1,\alpha=2\)
- we have
\[L({\lambda})=M({\lambda})/M({-\lambda-2})\] or \[0\to M({-\lambda-2})\to M({\lambda})\to L({\lambda})\to 0\]
- this gives a BGG resolution
- character of \(L({\lambda})\) = alternating sum of characters of Verma modules
\[\operatorname{ch}{L({\lambda})}=\operatorname{ch}{M({\lambda})}-\operatorname{ch}{M({-\lambda-2})}=\frac{e^{\lambda}}{1-e^{-2}}-\frac{e^{-\lambda-2}}{1-e^{-2}}\]
- comparison with Weyl-Kac character formula
\[\operatorname{ch} L({\lambda})=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\lambda+\rho)}}{e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})}=\frac{e^{\lambda+1}-e^{-\lambda-1}}{e^{1}(1-e^{-2})}\]
- In general, there are more terms involved in a BGG resolution and choosing right homomorphisms is not easy
- we take a detour
weak BGG resolution
- def
- We say that \(M \in O\) has a standard filtration (also called a Verma flag) if there is a sequence of submodules
\[0 = M_0 \subset M_1 \subset M_2 \subset \cdots \subset M_n = M\] for which each \(M^i := M_i/M_{i−1}\, (1 \le i \le n)\) is isomorphic to a Verma module.
- thm (Weak BGG resolution)
There is an exact sequence \[ 0 \to M({w_0\cdot \lambda}) = D_m^{\lambda} \to D_{m-1}^{\lambda} \to \cdots \to D_1^{\lambda} \to D_0^{\lambda}=M(\lambda) \to L(\lambda) \to 0 \] where \(D_{k}^{\lambda}\) has a standard filtration involving exactly once each of the Verma modules \(M(w\cdot \lambda)\) with \(\ell(w)=k\)
strategy to construct a BGG resolution
- construct a relative version of standard resoultion \(D_k:=U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\Lambda^{k}(\mathfrak{g}/\mathfrak{b})\) for \(L(0)\)
- construct a weak BGG resolution \(D_k^0:=D_k^{\chi_{0}}\) for \(L(0)\) by cutting down to the principal block component of each term
- construct a weak BGG resolution \(D_k^\lambda : = (D_k^0\otimes L(\lambda))^{\chi_{\lambda}}\) for \(L(\lambda)\) (we can also do this by applying the translation functor)
- show that it is actually a BGG resolution by computing \(\operatorname{Ext}\) between Verma modules
standard resolution of trivial module
- free \(U(\mathfrak{g})\)-modules \(U(\mathfrak{g})\otimes_{\mathbb{C}}\wedge^{k}(\mathfrak{g})\)
- standard resolution of trivial module in Lie algebra cohomology
\[\cdots \to U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{k}(\mathfrak{g})\to U(\mathfrak{g})\otimes_{\mathbb{C}}\wedge^{k-1}(\mathfrak{g})\to \cdots \to U(\mathfrak{g})\otimes_{\mathbb{C}}\wedge^{0}(\mathfrak{g})\to L(0)\]
- the sequence of modules \(D_k:=U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\wedge^{k}(\mathfrak{g}/\mathfrak{b})\) is a relative version of the standard resolution
- we can describe \(D_0\) and \(D_m\) explicitly
- define \(U(\mathfrak{g})\)-module homomorphism \(\partial_k : D_k \to D_{k-1}\) as
\[ \begin{align} \partial_k ( u\otimes \xi_1 \wedge \cdots \wedge \xi _k): &= \sum_{i=1}^k(-1)^{i+1}(uz_i\otimes \xi_1\wedge \cdots \wedge \hat{\xi_i}\wedge \cdots \wedge\xi_k)\\ &+\sum_{1\le i<j \le k} (-1)^{i+j}(u \otimes \overline{[z_iz_j]}\wedge \xi_1\wedge \cdots \wedge \hat{\xi_i}\wedge \cdots \wedge\hat{\xi_j} \wedge \cdots \xi_k) \end{align} \] where \(z_i\in \mathfrak{g}\) is a representative of \(\xi_i\in \mathfrak{g}/\mathfrak{b}\) and \(\overline{z}\) denotes the canonical surjection \(z\in\mathfrak{g}\) into \(\mathfrak{g}/\mathfrak{b}\).
- need to show that \(\partial_k\) is well-defined and it is actually a complex and exact
exactness
- exactness is tricky
- see Wallach, Real Reductive Groups I 6.A
- see Knapp, Lie Groups, Lie Algebras, and Cohomology IV.6
- Let \(U_j(\mathfrak{g}):=U^j(\mathfrak{g})U(\mathfrak{b})\) where \(U^j(\mathfrak{g})\) is the span of the PBW basis whose degree is \(\le j\)
- note that \(U_j(\mathfrak{g})=S_j(\mathfrak{n}^-)U(\mathfrak{b})\) where \(S_j(\mathfrak{n}^-)=\sum_{0\le k\le j}S^k(\mathfrak{n}^-)\).
- \(S^k(\mathfrak{n}^-)\) denote the elements of \(\operatorname{Sym}(\mathfrak{n}^-)\) that are homogeneous of degree \(k\).
- \(E_{j,k}:=U_j(\mathfrak{g})\otimes_{U(\mathfrak{b})}\wedge^{k}(\mathfrak{g}/\mathfrak{b})\)
- \(\{E_{j,k}\}_{j\ge 0}\) gives a filtration of \(D_k\)
- let \(\partial_0 : D_0\to L(0)\) be the canonical surjection
- exercise \[\partial_k : E_{j,k}\to E_{j+1,k-1}, k\ge 1\]
- \(\partial_k, k\ge 1\) induces \(\overline{\partial_k} : E_{j,k}/E_{j-1,k}\to E_{j+1,k-1}/E_{j,k-1}\)
- prop
For each \(j\ge 1\), \(\{(E_{j,k}/E_{j-1,k},\overline{\partial_k})\}_{-1\le k\le m+1}\) is an exact sequence. (here \(-1\) and \(m+1\) terms are zero)
- proof
As a vector space, \(E_{j,k}/E_{j-1,k}\cong S^j(\mathfrak{n}^-)\otimes \wedge^{k}(\mathfrak{n}^-)\).
Now we can apply basic results on the Koszul complexes :
- theorem
For each \(j\geq 1\), the following is exact \[ 0\to S^{j-m}(V)\otimes \wedge^m(V) \to S^{j-m+1}(V)\otimes \wedge^{m-1}(V) \to \cdots \to S^{j-1}(V)\otimes \wedge^{1}(V) \to S^{j}(V)\to 0 \] where \(S^j(V)=0\) for \(j<0\)
- see Lang, Algebra ' Koszul complex'
- examples
\[ 0\to \wedge^2 \to S^1\otimes \wedge^1 \to S^2\to 0 \] or \[ 0\to E_{0,2} \to E_{1,1}/E_{0,1} \to E_{2,0}/E_{1,0} \to 0 \]
\[ 0\to \wedge^1 \to S^1 \to 0 \] or \[ 0\to E_{0,1} \to E_{1,0}/E_{0,0} \to 0 \]
- in fact, we also have
\[ 0\to E_{0,0}\to L(0)\to 0 \]
- prop
For each \(j\ge 1\), \(\{(E_{j,k},\partial_k)\}_{-1\le k \le m+1}\) is exact. (here \(-1\) and \(m+1\) terms are zero)
- proof
Suppose \(u\in E_{j,k},\, j,k\geq 1\) satisfies \(\partial(u)=0\) in \(E_{j+1,k-1}\).
show : there exists \(v\in E_{j-1,k+1}\) such that \(\partial(v)=u\) in \[ E_{j-1,k+1} \to E_{j,k}\to E_{j+1,k-1}. \]
We look at \[ E_{j-1,k+1}/E_{j-2,k+1} \to E_{j,k}/E_{j-1,k} \to E_{j+1,k-1}/E_{j,k}. \] As \(\overline{\partial}(\overline{u})=0\), we can find \(v_1\in E_{j-1,k+1}\) such that \(\overline{\partial}(\overline{v_1})=\overline{u}\).
As \(\overline{u}-\overline{\partial}(\overline{v_1})=0\) in \(E_{j,k,}/E_{j-1,k}\), there exists \(w\in E_{j-1,k}\) such that \(u-\partial(v_1)=w\).
Now \(\partial(w)=0\) in \(E_{j,k-1}\) and by the same argument applied to \[ E_{j-2,k+1} \to E_{j-1,k}\to E_{j,k-1}, \] there exists \(v_2\in E_{j-2,k+1}\) such that \(w-\partial (v_2)\in E_{j-2,k}\), i.e. \(u-\partial(v_1)-\partial(v_2)\ \in E_{j-2,k}\).
By repeating this, we can find \(v_1,\cdots, v_j\) such that each \(v_i\in E_{j-i,k}\) and \(u-\partial(v_1)-\cdots -\partial(v_j)\ \in E_{0,k}\).
As \(\overline{\partial} : E_{0,k}\to E_{1,k-1}/E_{0,k-1}\) is injective and \(\partial(u-\partial(v_1)-\cdots -\partial(v_j))=0 \in E_{1,k-1}\), we can conclude \(u-\partial(v_1)-\cdots -\partial(v_j)=0\) or, \[ u=\partial(v_1)+\cdots +\partial(v_j). \] Thus if we set \(v:=v_1+\cdots +v_j\in E_{j-1,k+1}\) and \(\partial(v)=u\). ■
- thm
\(0\to D_m \to \cdots \to D_k\to \cdots \to D_1 \to D_0 \to L(0)\to 0\) is exact.
- proof
The above proposition clearly proves the exactness at \(D_k,\, k\ge 1\).
Assume that \(u\in D_0\), hence \(u\in E_{j,0}\) for some \(j\). Let \(u=u_0+u_1\) where \(u_0\) is the degree zero piece and \(u_1\) other terms in PBW basis.
If \(\partial(u)=0\), then it implies \(u_0 = 0\) as only degree zero term survives under \(\partial\).
Therefore \(j\ge 1\) and the above proposition still applies to conclude that there exists \(v\in E_{j-1,1}\) such that \(\partial(v)=u\).
This proves the exactness at \(D_0\). ■
weak BGG resolution of \(L(0)\)
- goal : find standard filtrations of \(D_k\) and \(D_k^0\) and their Verma subquotients
- lemma
Let \(N\) be a finite-dimensional \(U(\mathfrak{b})\)-module having a basis of weight vectors. Then \(M=U(\mathfrak{g})\otimes_{U(\mathfrak{b})} N\) has a standard filtration and each weight of \(N\) gives a corresponding Verma subquotient.
- proof
Let \(\{v_1,\cdots, v_r \}\) be a basis of \(N\) consisting of weight vectors and let \(\mu_i\) be the weight of \(v_i\).
We order the basis so that \(i\le j\) whenever \(\mu_i\le \mu_j\)
Let \(N_k\) be a space spanned by \(\{v_k,\cdots, v_r \}\) for \(1\le k \le r\).
exercise. Check that each \(N_k\) is a \(U(\mathfrak{b})\)-submodule. (hint : weight cannot decrease under \(U(\mathfrak{b})\) action)
We have a flag of \(U(\mathfrak{b})\)-modules : \[ 0 \subset N_r \subset N_{r-1} \subset \cdots \subset N_1 = N \label{Nflag} \]
We get a standard filtration of \(M\) from \ref{Nflag} as the functor \(N\mapsto U(\mathfrak{g})\otimes_{U(\mathfrak{b})} N\) is exact. (See Remark 1.3) ■
- this lemma is true even if we drop the assumption about the existence of basis of weight vectors
- but such module induces a module not necessarily on the BGG category
- prop
\(D_k\) has a standard filtration with Verma subquotients associated to sums of \(k\) distinct negative roots.
- proof
If we apply the above lemma, enough to answer :
Q. what are the weights of \(\wedge^k (\mathfrak{g}/\mathfrak{b})\) as \(\mathfrak{b}\)-module?
A : sum of \(k\) distinct negative roots ■
- prop
\(D_k^0\) has a standard filtration with Verma subquotients \(M(w\cdot 0), w\in W^{(k)}\) where \(W^{(k)}:=\{w\in W|\ell(w)=k\}\)
- proof
Taking a block preserves exactness :
If \(0\to M' \to M \to M(\mu)\to 0\), then \(0\to (M')^0 \to M^0 \to (M(\mu))^0 \to 0\).
\[ (M(\mu))^0 = \begin{cases} M(\mu) , & \text{if \]\mu\( is linked to \)0\( (\)\mu=w\cdot 0\( for some \)w\in W\()}\\ 0, & \text{otherwise} \\ \end{cases} \)
Thus we obtain a standard filtration of \(D_k^0\) from that of \(D_k\).
What are the Verma subquotients or when is \(\beta\) linked to \(0\) if \(\beta\) is given a sum of \(k\) distinct negative roots?
fact \[\ell(w)=|w \Phi^+ \cap \Phi^-|\].
exercise : Let \(\beta_w:=w\cdot 0\) for each \(w\in W\). Then \(\beta_w\) is a sum of elements in \(w \Phi^+ \cap \Phi^-\).
exercise : Let \(\Pi \subset \Phi^-\) be given. If the sum \(\beta\) of elements of \(\Pi\) is \(\beta_w\) for some \(w\in W\), then \(\Pi = w \Phi^+ \cap \Phi^-\). (we know the whole set by only looking at the sum of them)
Thus \(\beta\) is a sum of \(k\) distinct negative roots and linked to \(0\) iff there exists \(w\in W^{(k)}\).
Finally,
fact \[|W\cdot 0|=|W|\]. (this implies each \(\beta_w\) is distinct)
Therefore each \(M(w\cdot 0),\, w\in W^{(k)}\) appears only in once our standard filtration.
■
- thus we have found a weak BGG resolution of \(L(0)\)
weak BGG resolution of \(L(\lambda)\)
- based on Remark in 6.2
- let \(\lambda\in \Lambda^+\)
- goal : find a standard filtration of \(D_k^\lambda : = (D_k^0\otimes L(\lambda))^{\chi_{\lambda}}\) and its Verma subquotients
- prop
\(D_k^0\otimes L(\lambda)\) has a standard filtration with Verma subquotients \(M(w\cdot 0 + \mu)\) where \(w\in W^{(k)}\) and \(\mu\) is a weight of \(L(\lambda)\).
- proof
Use the following :
- thm (3.6)
Let \(M\) be a finite dimensional \(U(\mathfrak{g})\)-module. For any \(\lambda\in \mathfrak{h}^{*}\), \(T:=M(\lambda)\otimes M\) has a standard filtration with Verma subquotients \(M(\lambda+\mu)\). Here \(\mu\) ranges over the weights of \(M\), each occurring \(\dim M_{\mu}\) times in the filtration.
Tensoring with a finite-dimensional representation is an exact functor in BGG category (thm 1.1).
If \(0\to N\to M \to M(\lambda)\to 0\), then \(0\to N\otimes L(\lambda)\to M\otimes L(\lambda) \to M(\lambda)\otimes L(\lambda)\to 0\)
Use this to construct a standard filtration on \(D_k^0 \otimes L(\lambda)\) from that of \(D_k^0\). ■
- prop
\(D_k^\lambda\) has a standard filtration with Verma subquotients \(M(w\cdot \lambda),\, w\in W^{k}\).
- proof
Again taking the block component for \(\chi_{\lambda}\) is exact and it gives a standard filtration of \(D_k^\lambda\) from that of \(D_k^0\otimes L(\lambda)\).
We need to determine when \(w\cdot 0 + \mu\) is linked to \(\lambda\).
exercise \[w\cdot 0 + \mu\] is linked to \(\lambda\) iff \(\mu = w\lambda\).
As \(\lambda\) is the highest weight in \(L(\lambda)\), each \(w\lambda\) is with weight multiplicity 1. Thus each \(M(w\cdot \lambda),\, w\in W^{(k)}\) appears only once in our standard filtration. Note that each \(w\cdot \lambda\) is distinct as :
fact \[|W\cdot \lambda|=|W|\] for \(\lambda\in \Lambda^+\).■
extensions of Verma modules
- we have constructed a weak BGG resolution of \(L(\lambda)\) involving \(D_k^{\lambda}\)
- goal \[D_k^{\lambda}\] is a direct sum of Verma modules
- def
Let \(\mu, \lambda\in \mathfrak{h}^{*}\). Write \(\mu \uparrow \lambda\) if \(\mu = \lambda\) or there exists \(\alpha\in \Phi^+\) such that \(\mu=s_{\alpha}\cdot \lambda < \lambda \) (\(\mathbb{Z}^+\)-linear combination of simple roots)
We say \(\mu\) is strongly linked to \(\lambda\) if \(\mu = \lambda\) or there exist \(\alpha_1,\cdots, \alpha_r\in \Phi^+\) such that \(\mu=(s_{\alpha_1}\cdots s_{\alpha_r})\cdot \lambda \uparrow (s_{\alpha_2}\cdots s_{\alpha_r})\cdot \lambda \uparrow \cdots \uparrow ( s_{\alpha_r})\cdot \lambda \uparrow \lambda \)
- def
Let \(w,w'\in W\). Write \(w'\xrightarrow{t} w\) whenever \(w = t w' \) for some reflection \(t\in W\) and \(\ell(w') < \ell(w)\). Define \(w'<w\) if there is a sequence \(w'=w_0\to w_1\to \cdots \to w_n=w\). Extend this relation to a partial ordering of \(W\) and call it the Bruhat ordering.
example : http://groupprops.subwiki.org/wiki/File:Bruhatons3.png draw the edge of only the difference of length is 1.
exercise : Let \(w\in W, \alpha \in \Phi^+\) be given. The following are equivalent :
(i) There exists a \(\lambda\in \Lambda^{+}\) such that \(s_{\alpha}\cdot (w\cdot \lambda) \uparrow w\cdot \lambda\)
(ii) \(s_{\alpha}w > w\)
hint : use
fact \[w^{-1}\alpha>0\] iff \(\ell(s_{\alpha}w)> \ell(w)\).
- thm
(a) Let \(\lambda,\mu\in \mathfrak{h}^{*}\). If \(\operatorname{Ext}_{\mathcal{O}}(M(\mu),M(\lambda))\neq 0\), then \(\mu\) is strongly linked to \(\lambda\) but \(\mu \neq \lambda\)
(b) Let \(\lambda\in \Lambda^{+}\) and \(w,w'\in W\). If \(\operatorname{Ext}_{\mathcal{O}}(M(w'\cdot\lambda),M(w\cdot\lambda))\neq 0\), then \(w<w'\) in the Bruhat ordering. In particular, \(\ell(w)<\ell(w')\).
- proof of (a)
(a) uses projective cover, BGG reciprocity and BGG theorem from the previous chapters. (so we skip the proof) ■
- proof of (b)
From (a), we see that \(w'\cdot\lambda\) is strongly linked to \(w\cdot\lambda\). Thus we can find \( w'\cdot\lambda=(s_{\alpha_1}\cdots s_{\alpha_r})\cdot (w\cdot\lambda) \uparrow (s_{\alpha_2}\cdots s_{\alpha_r})\cdot (w\cdot\lambda) \uparrow \cdots \uparrow ( s_{\alpha_r})\cdot (w\cdot\lambda) \uparrow (w\cdot\lambda) \) From exercise, \(w'=s_{\alpha_1}\cdots s_{\alpha_r}w> \cdots > s_{\alpha_r}w > w\). ■
finish
- prop
A weak BGG resolution is a BGG resolution.
- proof
Use induction on the length of standard filtration.
Let \(M\subset D_k^{\lambda}\) and \(D_k^{\lambda}/M \cong M(w'\cdot \lambda)\) for some \(w'\in W^{(k)}\).
By induction hypothesis, \(0\to M=\oplus_{w\in W^{(k)},w\neq w'} M(w\cdot \lambda)\to D_k^{\lambda} \to M(w'\cdot \lambda) \to 0\), then this splits as \(\operatorname{Ext}\left(M(w'\cdot \lambda ),\oplus M(w\cdot \lambda)\right)\) is zero (ext is additive).
■
memo
- proof of Thm 3.6 uses the following (we don't need this for our goal)
- thm Tensor Identity (56p)
Let \(M\) be a \(U(\mathfrak{g})\)-module and \(L\) a \(U(\mathfrak{b})\)-module. Then \[ (U(\mathfrak{g})\otimes_{U(\mathfrak{b})}L)\otimes M \cong U(\mathfrak{g})\otimes_{U(\mathfrak{b})}(L \otimes M) \]
- prop (3.7)
Let \(M\in \mathcal{O}\) have a standard filtration. If \(\lambda\) is maximal among the weights of \(M\), then \(M\) has a submodule isomorphic to \(M(\lambda)\) and \(M/M(\lambda)\) has a standard filtration.
overview
- principal block : filtering through central characters
- is a block a \(U(\mathfrak{g})\)-submodule? yes
- how to check that it preserves the exactness : any homomorphism between modules belonging to different blocks will be zero
- how to describe \(\chi_{\lambda}\)? use the twisted Harish-Chandra homomorphism \(\psi : Z(\mathfrak{g})\to S(\mathfrak{h})\). we have
\[ \chi_{\lambda}(z)=(\lambda+\rho)(\psi(z)),\quad z\in Z(\mathfrak{g}) \]
- see 26p for an example of \(\chi_{\lambda}\) in type \(A_1\)
- combinatorial results
- longest elements satisfies \(w_0\cdot 0 = -2\rho\) (related to diagram automorphism)
- consider the set of sum of k distinct roots. Which elements are linked to \(0\)?
- Bruhat ordering
- Bruhat ordering and strong linkage relation
- let \(\lambda \in \Lambda^+\) (which is regular for the dot-action of \(W\))
- \(w'\cdot \lambda< w \cdot \lambda \) translates into \(w < w'\) in the Bruhat ordering
- strong linkage relation and extension of Verma modules
- for exterior powers, see Lie Algebras of Finite and Affine Type by Carter