Homology of invariants of a Weyl algebra under a finite group action (Q2701750)

Let \(\text{Sp}(2n,\mathbb{C})\) be the symplectic group and let \(G<\text{Sp}(2n,\mathbb{C})\) be a finite subgroup. The group \(G\) acts by automorphisms on the polynomial algebra \(\mathbb{C}[X_1,\dots,X_n,Y_1,\dots,Y_n]\) and on the Weyl algebra \(A_n(\mathbb{C})=\mathbb{C}[p_1,\dots,p_n,q_1,\dots,q_n]\) such that: \([p_i,p_j]=[p_i,q_j]=[q_i,q_j]=0\) if \(i\neq j\), \([p_i,q_i]= 1\).NEWLINENEWLINENEWLINEThe authors prove the following result which is a computation of the Hochschild homology and cohomology groups of the invariant algebra \(A_n(\mathbb{C})^G\): NEWLINE\[NEWLINE\dim_\mathbb{C} HH_j(A_n(\mathbb{C})^G)=\dim_\mathbb{C} H^{2n-j}(A_n(\mathbb{C})^G,A_n(\mathbb{C})^G)=a_j(G),NEWLINE\]NEWLINE where \(a_j(G)\), \(j\in\mathbb{N}\), denotes the number of conjugacy classes of elements of \(G\) having \(1\) as eigenvalue with multiplicity \(j\).