Kazhdan-Lusztig Polynomials, Subsingular Vectors and Conditionally Invariant (q-Deformed) Equations

Recently, there was a lot of interest in the study and applications in mathematics and physics of the singular [1] and subsingular vectors [2] of Verma modules. In particular, were considered the singular vectors of Verma modules of: the Virasoro algebra [3–7] the super-Virasoro algebras with N = 1 [8], N = 2 [9, 10] Kac-Moody algebras [11-13] quantum groups [14, 15] W-algebras [16,17] of Fock modules of the Virasoro algebra [18]. The physical applications are mostly in two-dimensional (super) conformal field theory, topological field theory, Calogero-Sutherland model, etc. Subsingular vectors figured prominently (though without explicit formulae) in the BRST analysis of the Fock modules of the Virasoro [19] and sl(2) Kac-Moody [20] algebras. Our interest in (sub)singular vectors is motivated by their relation to (conditionally) invariant equations. This relation stated in condensed form is: to every singular, resp., subsingular vector of a Verma module over a semi-simple (also reductive) Lie algebra G there corresponds a differential operator and equation invariant, resp., conditionally invariant with respect to G, cf. [21, 22]. Both statements are valid for the corresponding Drinfel’d-Jimbo quantum group U q (G), cf. [22, 23] and also for the corresponding Lie group (with some additional subtleties [21]).