# A question on the construction of finite W-algebras

In a well known construction of finite W-algebras, one first constructs a certain nilpotent subalgebra $\mathfrak{m}$ along with a character $\chi:\mathfrak{m}\rightarrow \mathbb{C}$. Then one defines

$$U(\mathfrak{g},e)=(U(\mathfrak{g})/U(\mathfrak{g})\mathfrak{m}_{\chi})^\mathfrak{m}$$

where $\mathfrak{m}_\chi$ is the set of all $m-\chi(m)$ and $\mathfrak{m}$ acts on $U(\mathfrak{g})$ by derivations, extending the adjoint action on $\mathfrak{g}$. Is this the same as

$$U(\mathfrak{g})^{\mathfrak{m}}/(U(\mathfrak{g})\mathfrak{m}_{\chi})^\mathfrak{m}?$$

Of course one can reformulate this question and ask if the following cohomology group vanishes:
$$H^1(\mathfrak{m},U(\mathfrak{g})\mathfrak{m}_{\chi})=0?$$ Maybe this follows from some Lynch style vanishing, but I am not very familiar with these theorems.

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Two very minor edits. Maybe lie-algebras would also be a useful tag? –  Jim Humphreys Sep 22 '10 at 16:46
Thanks, added lie-algebras tag. –  Jan Weidner Sep 22 '10 at 18:31

Thanks, unfortunately I do not yet quite see, how to use this in my situation. Do you suggest to show that $U(g)mχ=\mathbb{C}[M]\otimes_\mathbb{C} W$ for some trivial rep $W$ and then conclude $H^i(m,U(g)mχ)=H^i(m,\mathbb{C}[M])\otimes W=0$? Or did I get this completely wrong? –  Jan Weidner Sep 23 '10 at 9:01
I already read this paper, quite thoroughly. What Gan and Ginzburg show in these propositions is that the higher cohomology groups $H^i(m,U(g)/U(g)m_\chi)$ vanish. What I want is that $H^i(M,U(g)m_\chi)$ vanishes. I don't see how their propositions imply my statement, nor how to adapt their proof. Maybe I miss something obvious. –  Jan Weidner Sep 24 '10 at 7:40
Their proof goes as follows. By a spectral sequence argument they show that it is enough to check that $H^i(m,gr(U(g)/U(g)m_\chi))$ vanishes. Now they already showed that there is an equivariant isomorphism $gr(U(g)/U(g)m_\chi)=\mathbb{C}[M]\otimes \mathbb{C}[S]$ where the action on the second factor is trivial. Then they deduce $H^i(m,gr(U(g)/U(g)m_\chi))=H^i(m,\mathbb{C}[M])\otimes \mathbb{C}[S]$. Now as you already remarked $H^i(m,\mathbb{C}[M])$ is the algebraic deRham cohomology of $N\cong\mathbb(A)^n$, which vanishes for $i>0$. –  Jan Weidner Sep 24 '10 at 7:43