By J. Bernstein

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It is easy to check that Z(A) is a (commutative) subalgebra of A. 8. The centre of the group algebra k[G] is the space of class functions Ck (G). Proof. eh = egh . eg for all g ∈ G. But since eg is invertible with inverse eg−1 this is equivalent to the condition eg zeg−1 = z for every g ∈ G. Now since eg ek eg−1 = egkg−1 this is equivalent to λk egkg−1 = k∈G λ k ek , ∀g ∈ G. k∈G Thus z ∈ Z(A) if and only if λk = λgkg−1 for all g ∈ G, that is, if and only if z is a class function. 9. Let (V, ρ) be an irreducible representation of G, and let f ∈ Ck (G).

In the language we have just built up, our proof of Maschke’s theorem took an arbitrary projection from V to the subrepresentation W and applied IG to it (as an element of the representation Hom(V, V )) to get an equivariant projection. We are now finally able to make our irreducibility criterion more explicit: given a representation V , we wish to calculate dim(HomG (V, V )). By what we have shown above is that this dimension is just the rank of the projection operator IG on the representation Hom(V, V ).

Then V ⊗ V has a natural automorphism τ : V ⊗ V → V ⊗ V , given by τ (v ⊗ w) = w ⊗ v for all v, w ∈ V . Notice that since g ∈ G acts via g(v ⊗ w) = g(v) ⊗ g(w), the map τ is G-equivariant: τ (g(v ⊗ w)) = τ (g(v) ⊗ g(w)) = g(w) ⊗ g(v) = g(w ⊗ v) = g(τ (v ⊗ w)). It follows that G must preserve the eigenspaces of τ , and since τ 2 = idV , these are ±1. Let Λ2 (V ) to be the −1 eigenspace, and S 2 (V ) to be the +1 eigenspace. If {e1 , e2 , . . , en } is a basis for V , then it is not hard to see that the set of vectors {ei ⊗ ei , ei ⊗ ej + ej ⊗ ei : 1 ≤ i < j ≤ n} is a basis of S 2 (V ), while the set {ei ⊗ ej − ej ⊗ ei : 1 ≤ i < j ≤ n} 2 is a basis of Λ (V ).

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Algebraic Theory of D-modules [Lecture notes] by J. Bernstein


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