Whittaker modules for generalized Weyl algebras

Abstract

Abstract. We investigate Whittaker modules for generalized Weyl algebras, a class of associative algebras which includes the quantum plane, Weyl algebras, the universal enveloping algebra of sl 2 and of Heisenberg Lie algebras, Smith's generalizations of U (sl 2 ), various quantum analogues of these algebras, and many others. We show that the Whittaker modules V = Aw of the generalized Weyl algebra A = R(φ, t) are in bijection with the φ-stable left ideals of R. We determine the annihilator Ann A (w) of the cyclic generator w of V . We also describe the annihilator ideal Ann A (V ) under certain assumptions that hold for most of the examples mentioned above. As one special case, we recover Kostant's well-known results on Whittaker modules and their associated annihilators for U (sl 2 ).

Whittaker Modules for the Virasoro Algebra

“…U q (g).) Whittaker modules have also been studied for generalized Weyl algebras (see [3]) and in connection to non-twisted affine Lie algebras (see [5]). …”

Whittaker modules for a Lie algebra of Block type

“…Applying the PBW theorem, it is easy to see that {x λ w | λ ∈ P(K \ R)} is a basis of M ϕ over S(Z), where Z = Cx (1,0) .…”

“…Let A = S(Z), where Z = Cx (1,0) , and let I be an ideal of A. Write M = M ϕ , P 0 = P(K \ R), and w = p I w ∈ M/IM.…”

“…In this section, we characterize the Whittaker vectors in Whittaker modules for B, where ϕ is a fixed nonsingular Lie algebra homomorphism from n → C. Let M = M ϕ , w = 1 ⊗ 1, and Z = Z(B) = Cx (1,0) , the center of B. The notations in Section 3.1 will apply here.…”

“…In the quantum setting, Whittaker modules have been studied by Sevoystanov for U h (g) [8] and by Ondrus for U q (sl 2 ) [6]. Recently, Whittaker modules have also been studied by Ondrus and Wiesner for the Virasoro algebra [7], Zhang and Tan for Schrödinger-Virasoro algebra [17], Christodoulopoulou for Heisenberg algebras [2], and by Benkart and Ondrus for generalized Weyl algebras [1].…”