By Joseph Bernstein
The equivariant derived classification of sheaves is brought. All traditional functors on sheaves are prolonged to the equivariant scenario. a few purposes to the equivariant intersection cohomology are given. the idea could be invaluable to experts in illustration thought, algebraic geometry or topology.
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Extra resources for Equivariant Sheaves and Functors
Namely, we set 1 F (L) = F(L) meas(K /L) 2 (morally, one should think of elements of F (L) as half-measures on K /L). Note that the space F (L) has a canonical Hermitian metric. Namely, for φ1 , φ2 ∈ F (L) the product φ1 φ2 descends to a measure on K /L, which then can be integrated. Deﬁnition. Lagrangian subgroups L 1 , L 2 ⊂ K equipped with liftings to H , are called compatible if L 1 + L 2 is a closed subgroup and their lifting homomorphisms to H agree on L 1 ∩ L 2 . Note that for compatible subgroups one has L 1 + L 2 = (L 1 ∩ L 2 )⊥ .
1. 1. Fourier Transform Let K be a locally compact abelian group, K be the Pointriagin dual group. As was already mentioned earlier, the reader will not lose much by assuming that K is either a real vector space or a ﬁnite abelian group. For every Haar measure µ on K the Fourier transform is the operator Sµ : L 2 (K ) → L 2 ( K ) given by ˆ = Sµ (φ)(k) ˆ k(k)φ(k)µ. k∈K In this situation there is a unique dual measure µ ˆ on K such that Sµ is unitary with respect to the Hermitian metrics deﬁned using µ and µ.
5. 1) where v ∈ V , γ ∈ . Deﬁnition. 1). Its elements are called canonical theta functions for (H, , α). 2) coming from the Hermitian metric on L(H, α −1 ). 1) with the deﬁnition of the Fock representation we would like interpret the condition f ∈ T (H, , α) for a holomorphic function f on V as the invariance of f under operators Uα(γ ),γ for all γ ∈ (where is lifted to H using α). Since nonzero elements of T (H, , α) are not square-integrable, to achieve such an interpretation we have to extend operators Uλ,v to a larger space of holomorphic functions.
Equivariant Sheaves and Functors by Joseph Bernstein