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].
Geometric representation theory

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

maps between Verma modules

2 conditions to have non-zero homomorphisms \(V_{\lambda}\to V_{\mu}\) between two Verma modules
- \(\lambda+\rho, \mu+\rho\) are in the same orbit of Weyl group
- \(V_{\lambda}\leq V_{\mu}\), i.e. \(\lambda = \mu -\sum \alpha\), where the sum is over some positive roots.

example

SL2
- \(\lambda = \mu -2n\), \(n=0,1,2,\cdots\)
- \((\lambda+1)^2 = (\mu+1)^2\)

composition series of Verma modules

thm

The Verma module \(M(\lambda)\) has a finite composition series \[ M(\lambda)=N_0\supset N_1\supset N_2\supset \cdots N_{r}=O \] where each \(N_i\) is a submodule of \(M(\lambda)\) and \(N_{i+1}\) is a maximal submodule of \(N_i\). Moreover, \(N_i/N_{i+1}\) is isomorphic to \(L(w\cdot \lambda)\) for some \(w\in W\).

action of center on Verma modules

check

maximal submodule of Verma modules

Maximal Submodule of \(M(\lambda), \lambda \in \Lambda+\) (see 2.6 of Humphreys)

related items

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.

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Spacy 패턴 목록

[{'LOWER': 'verma'}, {'LEMMA': 'module'}]