By Alexander Polishchuk

This publication is a latest remedy of the speculation of theta capabilities within the context of algebraic geometry. the newness of its process lies within the systematic use of the Fourier-Mukai rework. Alexander Polishchuk starts off through discussing the classical conception of theta services from the perspective of the illustration thought of the Heisenberg workforce (in which the standard Fourier remodel performs the favourite role). He then indicates that during the algebraic method of this conception (originally as a result of Mumford) the Fourier-Mukai rework can frequently be used to simplify the present proofs or to supply thoroughly new proofs of many very important theorems. This incisive quantity is for graduate scholars and researchers with powerful curiosity in algebraic geometry.

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B) Generalize the proof of part (a) to show that for every n > 1 the space T (n H, , α1 · · · αn ) is spanned by the functions of the form f (v) = θ1 (v − a1 ) · · · θn (v − an ), where ai ∈ V , i ai = 0, θi ∈ T (H, , αi ). The classical (1-dimensional) theta series (with zero characteristics) is given by θ(z, τ ) = exp(πin 2 τ + 2πinz), n∈Z 36 6. Theta Functions I where z, τ ∈ C, Im(τ ) > 0. Let us consider the lattice = τ = Z + Zτ ⊂ C. Deﬁne the Hermitian form H = Hτ on C by H (z 1 , z 2 ) = [z 1 z 2 ]/[Im(τ )].

Such that the corresponding group K is ﬁnite). (a) Prove that there exists a Lagrangian subgroup I ⊂ K . (b) Let W be a representation of H such that U (1) acts by the identity character. Consider the decomposition W = ⊕χ∈ I Wχ 4. in isotipic components with respect to the I -action. Show that an element h ∈ H sends Wχ to Wχ−φh where φh is a character of I deﬁned by φh (i) = [h, i]. In the situation of the previous exercise show that the natural map W → F(I ) ⊗ W1 : w → (h → (hw)1 ) 26 5. 6. Representations of Heisenberg Groups I is an isomorphism of H -modules, where for w ∈ W we denote by w1 its projection to W1 .

One (Fock−∞ (V, J )) . has H 0 (T, L(H, α −1 )) T (H, , α) = Proof. The only fact one has to check is the inclusion T (H, , α) ⊂ Fock−∞ (V, J ). We leave this as an exercise for the reader. The space Fock−∞ (V, J ) is dual to the subspace Fock∞ (V, J ) ⊂ Fock(V, J ) consisting of f ∈ Fock(V, J ) such that f (x) = O x −n exp π H (x, x) 2 for all n. The subspace Fock∞ (V, J ) has the following representation-theoretic meaning: it is the space of C ∞ -vectors in the Fock representation. By the deﬁnition, these are vectors on which the action of the universal enveloping algebra of 24 Representations of Heisenberg Groups I the Lie algebra of H(V ) is well deﬁned.