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Almost-commuting variety, \(\mathcal{D}\)-modules, and Cherednik algebras. (English) Zbl 1158.14006

The authors study the almost-commuting variety and gave applications to the commuting variety, Cherednik algebras and quantum Hamiltonian reduction.
Let \(V=\mathbb{C}^n\) and \(\text{g}=\text{End}(V)=\text{gl}_n(\mathbb{C}).\) Write elements of \(V\) as column vectors, and elements of \(V^*\) as row vectors. Let \(\mathbb C[\text{g} \times \text{g} \times V \times V^*]=\mathbb C[X,Y,i,j]\) denote the polynomial algebra, and let \(\mathcal {J} \subset \mathbb C[X,Y,i,j]\) be the ideal generated by the \(n^2\) entries of the matrix \([X,Y]+ij.\) The almost-commuting variety is defined as \(\mathcal{M}=\text{Spec}\mathbb C[X,Y,i,j]/\mathcal{J}.\) For each integer \(k \in \{ 0,1,\dots,n \},\) let \[ \mathcal{M}'_k=\left\{ (X,Y,i,j) \in \mathcal{M} \;| \;\begin{matrix} Y \text{ has pairwise distinct eigenvalues,} \\ \dim(\mathbb C[X,Y]i)=n-k, \;\dim(j\mathbb C[X,Y])=k \end{matrix} \right\} \] and let \(\mathcal{M}_k\) be the closure of \(\mathcal{M}'_k\) in \(\mathcal{M}.\)
The main result is Theorem 1.1, which says: (1) \(\mathcal{M}\) is a complete intersection in \(\text{g} \times \text{g} \times V \times V^*;\) (2) the irreducible components of \(\mathcal{M}\) are \(\mathcal{M}_0,\dots,\mathcal{M}_n;\) (3) \(\mathcal{M}\) is reduced and equidimensional, of dimension \(\dim\mathcal{M}=n^2+2n.\) Some applications based on Theorem 1.1 are as follows: (1) The collection of pairs \((X,Y)\in{\text{g}}\times{\text{g}}\) with \([X,Y]=0\) forms the commuting variety \(Z\), which is an irreducible scheme. That whether or not \(Z\) is reduced is a long-standing open question. Let \(I\) be the vanishing ideal of \(Z\) and \(G=\text{GL}_n(\mathbb C).\) Theorem 1.3 says \(I^G=(\sqrt{I})^G,\) providing some evidence. (2) A rational Cherednik algebra (of \(\text{gl}_n\)-type) is determined by the tautological representation of the symmetric group \(S_n\) on \(\mathbb C^n\). (3) A study of \({ c}\)-twisted differential operators is included generalizing results from [loc. cit.].

MSC:

14A22 Noncommutative algebraic geometry
17B66 Lie algebras of vector fields and related (super) algebras
32C38 Sheaves of differential operators and their modules, \(D\)-modules
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