The group of automorphisms of the algebra of polynomial integro-differential operators

The group GnGn of automorphisms of the algebra In:=K〈x1,…,xn,∂∂x1,…,∂∂xn,∫1,…,∫n〉 of polynomial integro-differential operators is found:Gn=Sn⋉Tn⋉Inn(In)⊇Sn⋉Tn⋉GL∞(K)⋉⋯⋉GL∞(K)︸2n−1times,G1≃T1⋉GL∞(K), where SnSn is the symmetric group, TnTn is the n  -dimensional algebraic torus, Inn(In)Inn(In) is the group of inner automorphisms of InIn (which is huge). It is proved that each automorphism σ∈Gnσ∈Gn is uniquely determined by the elements σ(xi)σ(xi)ʼs or σ(∂∂xi)ʼs or σ(∫i)σ(∫i)ʼs. The stabilizers in GnGn of all the ideals of InIn are found, they are subgroups of finite   index in GnGn. It is shown that the group GnGn has trivial centre, InGn=K and InInn(In)=K, the (unique) maximal ideal of InIn is the only   nonzero prime GnGn-invariant ideal of InIn, and there are precisely n+2n+2GnGn-invariant ideals of InIn. For each automorphism σ∈Gnσ∈Gn, an explicit inversion formula   is given via the elements σ(∂∂xi) and σ(∫i)σ(∫i).