Let
U( \mathfrakg,e ) U\left( {\mathfrak{g},e} \right) be the finite W-algebra associated with a nilpotent element e in a complex simple Lie algebra
\mathfrakg = \textLie(G) \mathfrak{g} = {\text{Lie}}(G) and let I be a primitive ideal of the enveloping algebra
U( \mathfrakg ) 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 = \textAn\textnU( \mathfrakg )( Qe ?U( \mathfrakg,e )V ) 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( \mathfrakg,e ) U\left( {\mathfrak{g},e} \right) -module V, where Qe stands for the generalised Gelfand–Graev
\mathfrakg \mathfrak{g} -module associated with e. The main goal of this paper is to prove that the Goldie rank of the primitive quotient
U( \mathfrakg )
/
I {{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.} always divides dim V. For
\mathfrakg = \mathfraks\mathfrakln \mathfrak{g} = \mathfrak{s}{\mathfrak{l}_n} , we use a theorem of Joseph on Goldie fields of primitive quotients of
U( \mathfrakg ) U\left( \mathfrak{g} \right) to establish the equality
\textrk( U( \mathfrakg )
/
I ) = dimV {\text{rk}}\left( {{{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.}} \right) = \dim V . We show that this equality continues to hold for
\mathfrakg \ncong \mathfraks\mathfrakln \mathfrak{g} \ncong \mathfrak{s}{\mathfrak{l}_n} provided that the Goldie field of
U( \mathfrakg )
/
I {{{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. 相似文献
Letg be the Lie algebra of a connected reductive groupG over an algebraically closed field of characteristicp>0. Suppose thatG(1) is simply connected andp is good for the root system ofG. Ifp=2, suppose in addition thatg admits a nondegenerateG-invariant trace form. LetV be an irreducible and faithfulg-module withp-character g*. It is proved in the paper that dimV is divisible byp1/2dim() where () stands for the orbit of under the coadjoint action ofG.Oblatum 14-III-1994 & 17-XI-1994 相似文献
Let be an algebraic number field and be the ring of integers of . Let be a finite group and be a finitely generated torsion free -module. We say that is a globally irreducible -module if, for every maximal ideal of , the -module is irreducible, where stands for the residue field .
Answering a question of Pham Huu Tiep, we prove that the symmetric group does not have non-trivial globally irreducible modules. More precisely we establish that if is a globally irreducible -module, then is an -module of rank with the trivial or sign action of .
Let be an algebraically closed field of characteristic , a connected, reductive -group, , and the reduced enveloping algebra of associated with . Assume that is simply-connected, is good for and has a non-degenerate -invariant bilinear form. All blocks of having finite and tame representation type are determined.
Let G be a connected semisimple group over an algebraicallyclosed field K of characteristic p>0, and g=Lie (G). Fixa linear function g* and let Zg() denote the stabilizer of in g. Set Np(g)={xg|x[p]=0}. Let C(g) denote the category offinite-dimensional g-modules with p-character . In [7], Friedlanderand Parshall attached to each MOb(C(g)) a Zariski closed, conicalsubset Vg(M)Np(g) called the support variety of M. Suppose thatG is simply connected and p is not special for G, that is, p2if G has a component of type Bn, Cn or F4, and p3 if G has acomponent of type G2. It is proved in this paper that, for anynonzero MOb(C(g)), the support variety Vg(M) is contained inNp(g)Zg(). This allows one to simplify the proof of the KacWeisfeilerconjecture given in [18]. 相似文献
Let G be a connected reductive algebraic group over an algebraically closed field of characteristic p≥0. We give a case-free proof of Lusztig’s conjectures (Lusztig in Transform. Groups 10:449–487, 2005) on so-called unipotent pieces. This presents a uniform picture of the unipotent elements of G which can be viewed as an extension of the Dynkin–Kostant theory, but is valid without restriction on p. We also obtain analogous results for the adjoint action of G on its Lie algebra $\mathfrak{g}$ and the coadjoint action of G on $\mathfrak{g}^{*}$. 相似文献