A chiral Borel-Weil-Bott theorem (Q409631)

Let \(G\) be a simple complex Lie group, \({\mathfrak g}={\mathfrak n}_+\oplus{\mathfrak h}\oplus{\mathfrak n}_-\) its Lie algebra and \(X= G/({\mathfrak h}\oplus{\mathfrak n}_-)\) the corresponding flag manifold. Then there is a localization functor NEWLINE\[NEWLINE\Delta:{\mathfrak g}_-\text{Mod}\to{\mathcal D}^\lambda_X{_-\text{Mod}},NEWLINE\]NEWLINE where \({\mathcal D}^\lambda_X\) is the algebra of wisted differential operators acting on \({\mathcal L}_\lambda\); \({\mathcal L}_\lambda\) is the invertible \(G\)-equivariant sheaf of \({\mathcal O}_X\)-modules corresponding to an integral weight \(\lambda\in{\mathfrak h}^*\). Let \({\mathcal L}_{\nu_0}= \Delta(V_{\nu_0})\), \(V_{\nu_0}\) is the simple (finite-dimensional) \({\mathfrak g}\)-module with highest weight \(\nu_0\). Then the Borel-Weil-Bott theorem asserts \(H^0(X,{\mathcal L}_{\nu_0})= V_{\nu_0}\) and \(H^i(X,{\mathcal L}_{\nu_0})= 0\), \(i\geq 1\).NEWLINENEWLINE The authors and Chebotarev introduced an analogue of localization functor NEWLINE\[NEWLINE{\mathcal Z}hu_{\nu(z)}\circ \Delta:{\mathfrak g}_-\text{Mod}\to{\mathcal D}^{ch,tw}_X{_-\text{Mod}_{\nu(z)}},NEWLINE\]NEWLINE in the case of the affine Lie algebra \(\widehat{\mathfrak g}\), a universal central extension of \({\mathfrak g}\otimes \mathbb C[t, t^{-1}]\) [\textit{T. Arakawa}, \textit{D. Chebotarev} and \textit{F. Malikov}, Sel. Math., New Ser. 17, No. 1, 1--46 (2011; Zbl 1233.17021)], hereafter referred to as [1]. Here, \({\mathcal D}^{ch,tw}_X\) is the sheaf of twisted chiral differential operators proposed in [1], and \(\nu(z)= \nu_0/z+ \nu_{-1}+ \nu_{-2}z+\cdots\in{\mathfrak h}^*((z))\). \({\mathcal L}^{ch}_{\nu(z)}={\mathcal Z}hu_{\nu(z)}\circ\Delta(V_{\nu_0})\) is an analogue of \({\mathcal L}_{\nu_0}\) in the case of \(\widehat{\mathfrak g}\).NEWLINENEWLINE In this paper, formal character \(\text{ch\,}H^i(X,{\mathcal L}^{ch}_{\nu(z)})\) is defined (\S3,(3.12)) and set NEWLINE\[NEWLINE\chi({\mathcal L}^{ch}_{\nu(z)})= \sum^{\dim X}_{i=0} (-1)^i\text{ch\,}H^i(X,{\mathcal L}^{ch}_{\nu(z)}).NEWLINE\]NEWLINE Then the following chiralization of the Borel-Weil-Bott theorem is proved.NEWLINENEWLINE NEWLINETheorem 1.1. Let \(W\) be the Weyl group of \({\mathfrak g}\), \(\ell(w)\) is the length of \(w\in W\), then NEWLINE\[NEWLINE\chi({\mathcal L}^{ch}_{\nu(z)})= \sum_{w\in W} (-1)^{\ell(w)} e^{w\circ\nu_0}\times \prod_{w\in\widehat\Delta\, re_+} (1- e^{-\alpha})^{-1},NEWLINE\]NEWLINE NEWLINE\[NEWLINEH^i(X,{\mathcal L}^{ch}_{\nu(z)})= \bigoplus_{w\in W,\ell(w)= i} \mathbb V_{nu(z)}[\langle\nu_0- w\circ\nu_0, \rho^\nu\rangle].NEWLINE\]NEWLINE Here \(\mathbb V_{\nu(z)} [m]\) stands for \(\mathbb V_{nu(z)}\) as a \(\widehat{\mathfrak g}\)-module with conformal filtration shifted by \(m\) (explained in \S3.1.5).NEWLINENEWLINE NEWLINECorollary 1.2. NEWLINE\[NEWLINE\text{ch\,}\mathbb V_{\nu(z)}= {\sum_{w\in W}(-1)^{\ell(w)} e^{w\circ\nu_0}\over \prod_{\alpha\in\Delta_+} (1- e^{-\langle \nu_0+ \rho,\alpha^\vee\rangle\delta}) \prod_{\alpha\in \widehat\Delta^{re}_+} (1- e^{-\alpha})}.NEWLINE\]NEWLINE In the homogeneous grading specialization (\(e^\alpha\to 1\), \(e^{-\delta}\to q\)), we have simpler formulas NEWLINE\[NEWLINE\chi({\mathcal L}^{ch}_{\nu(z)}, q)= \dim V_{\nu_0} \prod^\infty_{j=1} (1- q^j)^{-2\dim X},NEWLINE\]NEWLINE NEWLINE\[NEWLINE\dim_Q\mathbb V_{\nu(z)}= \dim V_{\nu_0} \prod^\infty_{j=1} (1- q^j)^{-2\dim X} \prod_{\alpha\in\Delta_+} (1- q^{\langle\nu_0+ \rho, \alpha^\vee\rangle})^{-1}.NEWLINE\]NEWLINE The authors remark this character formula is not new [\textit{A. Arakawa}, Characters of representations of affine Kac-Moody-Lie algebras at the critical level, \url{arXiv:0706.1817}, \textit{E. Frenkel} and \textit{D. Gaitsgory}, Weyl modules and opers without monodromy. Arithmetic and geometry around quantization. Basel: Birkhäuser. Prog. Math. 279, 101--121 (2010; Zbl 1231.17007)], and say the point is not so much the formula itself but the fact that it nicely fits in and follows the proposed geometric framework.NEWLINENEWLINE NEWLINEThis character formula can be applied to the computation of elliptic genus of \(X\). Because introducing generating function NEWLINE\[NEWLINE\begin{multlined} {\mathcal E}_\lambda={\mathcal L}_\lambda\otimes \Biggl(\bigoplus^\infty_{n=1} \Biggl(\bigoplus^\infty_{m=0} q^{nm} S^m{\mathcal T}_X\Biggr)\Biggr)\otimes \Biggl(\bigotimes^\infty_{n=1} \Biggl(\bigotimes^\infty_{m=0} q^{nm} S^m \Omega_X\Biggr)\Biggr)=\\ {\mathcal L}_\lambda+q{\mathcal E}_{\lambda,1}+ q^2{\mathcal E}_{\lambda,2}+\cdots,\end{multlined}NEWLINE\]NEWLINE and define \(\chi({\mathcal E}_\lambda, q)= \chi({\mathcal L}_\lambda)+ q\chi({\mathcal E}_{\lambda, 1})+\cdots\), it is known \(\chi({\mathcal E}_\lambda, q)= \chi({\mathcal L}_\lambda)+ q\chi({\mathcal E}_{\lambda, 1})+\cdots\), is the elliptic genus \(g_Q(X, q)\) of \(X\) [\textit{L. Borisov} and \textit{A. Libgober}, Invent. Math. 140, No. 2, 453--485 (2000; Zbl 0958.14033)], where NEWLINE\[NEWLINEQ(x)= {x\over 1-e^{-x}} \prod^\infty_{n=1} (1- q^n e^{-x})^{-1}(1- q^n e^x)^{-1}.NEWLINE\]NEWLINE In [\textit{E. Frenkel} and \textit{D. Gaitsgory}, Local geometric Langlands correspondence: the spherical case. Algebraic analysis and around. Adv. Stud. Pure Math. 54, 167--186 (2009; Zbl 1192.17012)], it is shown NEWLINE\[NEWLINEH^{\infty/2+k}_{DS}(L{\mathfrak n}_+, H^i(X,{\mathcal L}^{ch}_{\nu(z)}))= \bigoplus^{m_i}_{j=1} \mathbb C[n_{ij}],NEWLINE\]NEWLINE if \(k=0\), and \(=0\) otherwise. It also follows from this paper \(H^i(X,{\mathcal L}^{ch}_{\nu(z)})= \bigoplus^{m_i}_{j=1} \mathbb V_{\nu(z)}[n_{ij}]\) (meaning of \(\mathbb V_{\nu(z)}[n_{ij}]\) is explained in \S4.3). Here the Drinfeld-Sokolov reduction functorNEWLINENEWLINE\(H^{\infty/2+\bullet}_{DS}(L{\mathfrak n}_+,M)\) is the cohomology of \((M\otimes C\ell({\mathfrak n}_+), d)\), \(C\ell({\mathfrak n}_+)\) is the vertex Clifford algebra defined in \S2.1.5 [cf. \textit{E. Frenkel} and \textit{D. Ben-Zvi}, Vertex algebras and algebraic curves. 2nd ed. Providence, RI: American Mathematical Society (2004; Zbl 1106.17035)]. Therefore main part of this paper is detailed studies of Drinfeld-Sokolov reduction at the critical level, which are given in \S3. They contain the following results, which are interesting in themselvesNEWLINENEWLINE 1. the functor \(H^{\infty/2+i}_{DS}(L{\mathfrak n}_+,?)= 0\) if \(i>0\);NEWLINENEWLINE 2. the functor \(H^{\infty/2+i}_{DS}(L{\mathfrak n}_+,?)\) is right exact, and the class of modules with Verma filtration is adapted to this functor;NEWLINENEWLINE 3. \(H^{\infty/2+0}_{DS}(L{\mathfrak n}_+,?)\), \(i>0\) is isomorphic to the derived functor \(L^i H^{\infty/2+0}_{DS}(L{\mathfrak n}_+,?)\),NEWLINENEWLINE (Th.3.5). The authors say this result shows somewhat unexpectedly, that the Drinfeld-Sokolov reduction, \(H^{\infty/2+\bullet}_{DS}(L{\mathfrak n}_+,?)\), is more like homology. Then Theorem 1.1 and Corollary 1.2 are proved in \S4.NEWLINENEWLINE Definitions and examples of vertex algebras and chiral differential operators, including definition of \({\mathcal Z}hu\), are given in \S2.