Double affine Hecke algebras and congruence groups

Abstract: The most general construction of double affine Artin groups (DAAG) and Hecke algebras (DAHA) associates such objects to pairs of compatible reductive group data. We show that DAAG/DAHA always admit a faithful action by automorphisms of a finite index subgroup of the Artin group of type $A_{2}$, which descends to a faithful outer action of a congruence subgroup of $SL(2,\mathbb{Z})$ or $PSL(2,\mathbb{Z})$. This was previously known only in some special cases and, to the best of our knowledge, not even conjectured to hold in full generality. The structural intricacies of DAAG/DAHA are captured by the underlying semisimple data and, to a large extent, by adjoint data; we prove our main result by reduction to the adjoint case. Adjoint DAAG/DAHA correspond in a natural way to affine Lie algebras, or more precisely to their affinized Weyl groups, which are the semi-direct products $W\ltimes Q^{\vee}$ of the Weyl group $W$ with the coroot lattice $Q^{\vee}$. We now describe our results for the adjoint case in greater detail. We first give a new Coxeter-type presentation for adjoint DAAG as quotients of the Coxeter braid groups associated to certain crystallographic diagrams that we call double affine Coxeter diagrams. As a consequence we show that the rank two Artin groups of type $A_{2},B_{2},G_{2}$ act by automorphisms on the adjoint DAAG/DAHA associated to affine Lie algebras of twist $r=1,2,3$, respectively. This extends a fundamental result of Cherednik for $r=1$. We show further that the above rank two Artin group action descends to an outer action of congruence subgroup $\Gamma_{1}(r)$. In particular $\Gamma_{1}( r) $ acts naturally on the set of isomorphism classes of representations of an adjoint DAAG/DAHA of twist type $r$, giving rise to a projective representation of $\Gamma_{1}( r) $ on the space of a $\Gamma_{1}( r) $-stable representation.