Verma modules

introduction

• The study of Verma modules was initiated by Verma [V] who showed that any nonzero homomorphism between Verma modules is injective and occurs with multiplicity one.
• He also found a sufficient condition for the existence of nontrivial homomorphism between Verma modules and conjectured that this condition is also necessary.
• The conjecture was ultimately proved by Bernstein-Gelfand-Gelfand [BGG1] who introduced the well-known category O to study representations of complex semismiple Lie algebras [BGG2].

infinite in both direction

• How to construct a representation with basis \(\{v_j|j\in \mathbb{Z}\}\)

brute force

• impose the following conditions

\[H v_j=c_j v_j\] \[F v_j=b_jv_{j+1}\] \[E v_j=a_jv_{j-1}\]

• we get the following conditions

$$ \begin{align} a_j b_{j-1}-a_{j+1} b_j+c_j=0 \\ a_j \left(c_{j-1}-c_j-2\right)=0\\ b_j \left(-c_j+c_{j+1}+2\right)=0 \end{align} $$

• Fix $c_j=\lambda-2j$. Then as long as $b_j a_{j+1}-b_{j-1} a_{j}=\lambda -2j$ is satisfied, we get a $U$-module structure on the space spanned by \(\{v_j|j\in \mathbb{Z}\}\)

symmetrical choice

\[H v_j=(\lambda -2j)v_j\] \[F v_j=(j-\frac{\lambda }{2})v_{j+1}\] \[E v_j=(\frac{\lambda }{2}-j)v_{j-1}\]

semi-infinite case : Verma module

• How to construct a representation $V(\lambda)$ with basis \(\{v_j|j\geq 0\}\)
• \(\lambda\in \mathbb{F}\) 에 대하여, highest weight vector \(v_0\) 를 정의

\[Ev_0=0\]\[Hv_0=\lambda v_0\]

• impose the following conditions

\[H v_j=(\lambda -2j)v_j\]\[F v_j=(j+1)v_{j+1}\]\[E v_j=(\lambda -j+1)v_{j-1}\]

finite representation

• \(\{v_j|j\geq 0\}\) 가 생성하는 벡터공간 $V(\lambda)$ 이 유한차원인 L-모듈이 되려면, \(\lambda\in\mathbb{Z}, \lambda\geq 0\) 이 만족되어야 한다

related items

• BGG category and BGG resolution
• Tilting modules for quantum groups

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxRUtEQlV4M2FxazQ/edit

articles

• Xiao, Wei. ‘Differential Equations and Singular Vectors in Verma Modules’. arXiv:1503.06385 [math], 22 March 2015. http://arxiv.org/abs/1503.06385.
• Xu, Xiaoping. ‘Differential-Operator Representations of $S_n$ and Singular Vectors in Verma Modules’. arXiv:0903.4239 [math], 25 March 2009. http://arxiv.org/abs/0903.4239.
• Fuchs, Dmitry, and Constance Wilmarth. ‘Projections of Singular Vectors of Verma Modules over Rank 2 Kac-Moody Lie Algebras’. Symmetry, Integrability and Geometry: Methods and Applications, 27 August 2008. doi:10.3842/SIGMA.2008.059.
• Chari, Vyjayanthi. ‘Annihilators of Verma Modules for Kac-Moody Lie Algebras’. Inventiones Mathematicae 81, no. 1 (1985): 47–58. doi:10.1007/BF01388771.
• Duflo, Michel. ‘Construction of Primitive Ideals in an Enveloping Algebra’. In Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), 77–93. Halsted, New York, 1975. http://www.ams.org/mathscinet-getitem?mr=0399194.
• Van den Hombergh, A. ‘Note on a Paper by Bernšteĭn, Gel'fand, and Gel'fand on Verma Modules’. Nederl. Akad. Wetensch. Proc. Ser. A 77, Indag. Math. 36 (1974): 352–56.
• Malikov, F. G., B. L. Feigin, and D. B. Fuks. ‘Singular Vectors in Verma Modules over Kac—Moody Algebras’. Functional Analysis and Its Applications 20, no. 2 (1 April 1986): 103–13. doi:10.1007/BF01077264.
• Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Differential Operators on the Base Affine Space and a Study of $\mathfrak{g}$-Modules’. In Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), 21–64. Halsted, New York, 1975. http://www.ams.org/mathscinet-getitem?mr=0578996.
• Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Structure of Representations That Are Generated by Vectors of Highest Weight’. Akademija Nauk SSSR. Funkcional\cprime Nyi Analiz I Ego Priloženija 5, no. 1 (1971): 1–9.
• Verma, Daya-Nand. ‘Structure of Certain Induced Representations of Complex Semisimple Lie Algebras’. Bulletin of the American Mathematical Society 74 (1968): 160–66.
• [V] Verma, Structure of certain induced representations of complex semisimple Lie algebras, Ph.D. thesis, Yale Univ. 1966.