We extend Lawrence's representations of the braid groups to relative homology modules, and we show that they are free modules over a Laurent polynomials ring. define homological operators actually provide representation for an integral version $U_q \mathfrak{sl}(2)$. suggest isomorphism between given basis standard tensor products Verma it preserve ring coefficients, action \mathfrak{sl}(2)$, g...

We show that bigrassmannian permutations determine the socle of cokernel an inclusion Verma modules in type $A$. All such socular constituents turn out to be indexed by Weyl group elements from penultimate two-sided cell. Combinatorially, a module $w\in S_n$ into dominant are shown determined essential set $w$ and their degrees graded picture computable terms associated rank function. As applic...

Let G be a p-adic connected reductive group with Lie algebra g. For a parabolic subgroup P ⊂ G and a finite-dimensional locally analytic representation V of a Levi subgroup of P, we study the induced locally analytic G-representation W = IndP (V ). Our result is the following criterion concerning the topological irreducibility of W : If the Verma module U(g) U(p) V ′ associated to the dual repr...

Let L be a finite-dimensional, semisimple Lie algebra over an algebraically closed field k of characteristic 0. Let H be a fixed Cartan subalgebra of L, and Φ be the root system. Fix a base ∆ = {α1, · · · , αl} of Φ. Let Λ denote the set of dominant, integral linear functions on H. Theorem 0.1. There is a one-to-one correspondence Λ ∼ −→ {isomorphism classes of finite-dimensional irreducible L-...

We construct a generator system of the annihilator of a generalized Verma module of a classical reductive Lie algebra induced from a character of a parabolic subalgebra as an analogue of the minimal polynomial of a matrix. In a classical limit it gives a generator system of the defining ideal of any semisimple co-adjoint orbit of the Lie algebra. We also give some applications to integral geome...

Let g be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S. For I ⊆ S let g I be the corresponding semi-simple subalgebra of g. Denote by W I the Weyl group of g I and let w • and w I • be the longest elements of W and W I , respectively. In this paper we show that the answer to Kostant's problem, i.e. whether the universal enveloping algebra surjec...

We prove a conjecture of Kuznetsov stating that the equivariant K-theory affine Laumon spaces is universal Verma module for quantum algebra U_q(gl_n^). do so by reinterpreting action toroidal U_q(gl_n^^) on from [14] in terms shuffle studied [12], which constructs an embedding U_q(gl_n^) into

Abstract. We consider a tensor product of a Verma module and the linear representation of sl(n + 1). We prove that the corresponding phase function, which is used in the solutions of the KZ equation with values in the tensor product, has a unique critical point and show that the Hessian of the logarithm of the phase function at this critical point equals the Shapovalov norm of the corresponding...

We prove an explicit formula for a projection of singular vectors in the Verma module over a rank 2 Kac-Moody Lie algebra onto the universal enveloping algebra of the Heisenberg Lie algebra and of sl2. The formula is derived from a more general but less explicit formula due to Feigin, Fuchs, and Malikov [3]. In the simpler case of A1 the formula was obtained in [4].