The Picard Group of a Noncommutative Algebraic Torus

We compute the Picard group $Pic(A_q)$ of the noncommutative algebraic 2-torus $A_q$, describe its action on the space $R(A_q)$ of isomorphism classes of rk 1 projective modules and classify the algebras Morita equivalent to $A_q$. Our computations are based on a quantum version of the Calogero-Moser correspondence relating projective $A_q$-modules to irreducible representations of the double affine Hecke algebras (DAHA) $H_{t, q^{-1/2}}(S_n)$ at $t = 1$. We show that, under this correspondence, the action of $Pic(A_q)$ on $R(A_q)$ agrees with the action of $SL_2(Z)$ on $H_{t, q^{-1/2}}(S_n)$ constructed by I.Cherednik. We compare our results with smooth and analytic cases. In particular, when $|q| \not= 1$, we find that $Pic(A_q)$ is isomorphic to the group of auto-equivalences $Auteq(D^b(X))/Z$ of the bounded derived category of coherent sheaves on the elliptic curve $X = C*/Z$ modulo translations.