caa a22 4500 445862300 CHVBK 20180317145447.0 cr unu---uuuuu 170323e20110901xx s 000 0 eng 10.1007/s00031-011-9141-x doi (NATIONALLICENCE)springer-10.1007/s00031-011-9141-x Premet Alexander School of Mathematics, The University of Manchester, Oxford Road, M13 9PL, Oxford, UK aut Enveloping algebras of Slodowy slices and Goldie rank [Elektronische Daten] [Alexander Premet] Let $ U\left( {\mathfrak{g},e} \right) $ be the finite W-algebra associated with a nilpotent element e in a complex simple Lie algebra $ \mathfrak{g} = {\text{Lie}}(G) $ and let I be a primitive ideal of the enveloping algebra $ U\left( \mathfrak{g} \right) $ whose associated variety equals the Zariski closure of the nilpotent orbit (Ad G) e. Then it is known that $ I = {\text{An}}{{\text{n}}_{U\left( \mathfrak{g} \right)}}\left( {{Q_e}{ \otimes_{U\left( {\mathfrak{g},e} \right)}}V} \right) $ for some finite dimensional irreducible $ U\left( {\mathfrak{g},e} \right) $ -module V, where Q e stands for the generalised Gelfand-Graev $ \mathfrak{g} $ -module associated with e. The main goal of this paper is to prove that the Goldie rank of the primitive quotient $ {{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.} $ always divides dim V. For $ \mathfrak{g} = \mathfrak{s}{\mathfrak{l}_n} $ , we use a theorem of Joseph on Goldie fields of primitive quotients of $ U\left( \mathfrak{g} \right) $ to establish the equality $ {\text{rk}}\left( {{{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.}} \right) = \dim V $ . We show that this equality continues to hold for $ \mathfrak{g} \ncong \mathfrak{s}{\mathfrak{l}_n} $ provided that the Goldie field of $ {{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.} $ is isomorphic to a Weyl skew-field and use this result to disprove Joseph's version of the Gelfand-Kirillov conjecture formulated in the mid-1970s. Springer Science+Business Media, LLC, 2011 Transformation Groups SP Birkhäuser Verlag Boston 16/3(2011-09-01), 857-888 1083-4362 16:3<857 2011 16 31 https://doi.org/10.1007/s00031-011-9141-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00031-011-9141-x text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Premet Alexander School of Mathematics, The University of Manchester, Oxford Road, M13 9PL, Oxford, UK aut NATIONALLICENCE 773 0- Transformation Groups SP Birkhäuser Verlag Boston 16/3(2011-09-01), 857-888 1083-4362 16:3<857 2011 16 31 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer