Harish-Chandra bimodules for quantized Slodowy slices

Abstract

Abstract. The Slodowy slice is an especially nice slice to a given nilpotent conjugacy class in a semisimple Lie algebra. Premet introduced noncommutative quantizations of the Poisson algebra of polynomial functions on the Slodowy slice.In this paper, we define and study Harish-Chandra bimodules over Premet's algebras. We apply the technique of Harish-Chandra bimodules to prove a conjecture of Premet concerning primitive ideals, to define projective functors, and to construct "noncommutative resolutions" of Slodowy slices via translation functors.

Multiplicity-Free Primitive Ideals Associated With Rigid Nilpotent Orbits

“…Therefore, the primitive ideals I(λ) and I(λ ′ ) are distinct and hence so are the respective one-dimensional representations of U(g, e). 0-roots and 1-roots are (42,18) and (48,20), respectively, and the total contribution of all roots is (45,19). The orbit O(e) is non-special in the present case and comparing its size with the size of a root subsystem of type A 8 in Φ indicates that one should again seek…”

“…We call VA(I) the associated variety of I and denote by X O the set of all I ∈ X with VA(I) = O. It is known that I V ∈ X O(e) for any finite dimensional irreducible U(g, e)-module V and any I ∈ X O(e) can be presented in this form for at least one such V ; see [37], [38], [30], [18], [39].…”

“…Numerical experiments indicate that e is quite immune to the choice of pinning and the following one is probably the best possible: (28,54,64,34,18) and (16,32,40,20,10), respectively. The total contribution of all roots is (22,43,52,27,14).…”

“…The roots are given below. The (2, 5)-contributions of the 0-roots and 1-roots are (16,28) and (18,27), respectively, and the total contribution of all roots is 34 2 , 55 2 . In the present case the orbit O(e) is non-special and comparing the size of O(e) with the size of a root subsystem of type D 5 + A 3 in Φ indicates that one should seek…”

“…The (1, 2, 3, 5, 7)-contributions of the 0-roots and 1-roots are (14,19,26,29,9) and (8,12,16,18,8), respectively, and the total contribution of all roots is (11, ). So Condition (C) for λ + ρ reads 2a 1 + 2a 2 + 3a 3 + 4a 4 + 3a 5 + 2a 6 + a 7 = 11 4a 1 + 7a 2 + 8a 3 + 12a 4 + 9a 5 + 6a 6 + 3a 7 = 31 3a 1 + 4a 2 + 6a 3 + 8a 4 + 6a 5 + 4a 6 + 2a 7 = 21 6a 1 + 9a 2 + 12a 3 + 18a 4 + 15a 5 + 10a 6 + 5a 7 = 47 2a 1 + 3a 2 + 4a 3 + 6a 4 + 5a 5 + 4a 6 + 3a 7 = 17.…”