Algebraic Geometry and Number Theory: In Honor of Vladimir by Alex Eskin, Andrei Okounkov (auth.), Victor Ginzburg (eds.)

By Alex Eskin, Andrei Okounkov (auth.), Victor Ginzburg (eds.)

One of the main inventive mathematicians of our occasions, Vladimir Drinfeld acquired the Fields Medal in 1990 for his groundbreaking contributions to the Langlands software and to the idea of quantum groups.

These ten unique articles by way of well-known mathematicians, devoted to Drinfeld at the party of his fiftieth birthday, generally mirror the variety of Drinfeld's personal pursuits in algebra, algebraic geometry, and quantity theory.

Contributors: A. Eskin, V.V. Fock, E. Frenkel, D. Gaitsgory, V. Ginzburg, A.B. Goncharov, E. Hrushovski, Y. Ihara, D. Kazhdan, M. Kisin, I. Krichever, G. Laumon, Yu.I. Manin, A. Okounkov, V. Schechtman, and M.A. Tsfasman.

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For example, if Cαβ = Cβα = −1, then Fα H α (x)Fβ Fα = H α (1 + x −1 )−1 H β (1 + x)Fβ H α (x)−1 Fα Fβ H α (1 + x)H β (1 + x −1 )−1 . 5 minus part 5 and the “map ev is Poisson’’ part Map 1 is a corollary of the obvious property αβ ¯ → β α. ¯ Map 2 follows from αβ → βα ¯ ¯ and α¯ β → β α. ¯ Map 6 follows from αα → α and α¯ α¯ → α. ¯ Each of these maps is obviously a composition of a mutation µ(αnα (A)+1 ) , or its α¯ version, and the projection along the corresponding coordinate. The maps 3 follow from αβα → βαβ and α¯ β¯ α¯ → β¯ α¯ β¯ and are given by the mutations µ(αnα (A)+1 ) , or its α¯ version.

The latter matrix elements are determined in the following. Proposition 1. We have ⎧ ⎪1, k ≡ l ≡ 0 mod 2, (Wek , el ) ⎨ = 0, k ≡ l ≡ 1 mod 2, ⎪ b(k)b(l) ⎩ 2/(k − l), otherwise, where b(k) = 2k k! 3 For the proof of Proposition 1, form the generating function f (x, y) = x k y l (Wek , el ). k,l From the equality exp n>0 x 2n+1 2n + 1 = 1+x 1−x and definitions, we compute f (x, y) = 1 1 − xy 1+x 1−x 1−y . 1+y The factorization x ∂ ∂ −y ∂x ∂y f (x, y) = (x + y)(1 + x)(1 − y) (1 − x 2 )3/2 (1 − y 2 )3/2 by elementary binomial coefficient manipulations proves (27) for k = l.

Quasimodular forms for form a graded algebra denoted by QM( ). By a theorem of Kaneko and Zagier [13], QM( ) = Q[E2 ] ⊗ M( ). In particular, E2 (q), E2 (q 2 ), E2 (q 4 ) ∈ QM( 0 (4)) (40) where 0 (4) ab cd = c ≡ 0 mod 4 ⊂ SL2 (Z). Hence all averages (10) lie in QM( 0 (4)). In fact, the series (40) generate the subalgebra QM2∗ ( 0 (4)) of even weight quasimodular forms. This is because M2∗ ( 0 (4)) is freely generated by two generators of weight two, for example, by E2odd (q) and E2odd (q 2 ), where ⎛ ⎞ ∞ 1 ⎝ + d ⎠ q n.

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