HAAR WEIGHT ON SOME QUANTUM GROUPS S. L. WORONOWICZ Abstract. We present a number of examples of locally compact quantum groups. These are quantum deformations of the group of affine transformations R (‘ax+b’ group) and C (Gz group). Starting from a modular multiplicative unitary W we find (under certain technical assumption) a simple formula expressing the (right) Haar weight on the quantum group associated with W . The formula works for quantum ‘ax + b’ and ‘az + b’ groups. 0. Introduction It is difficult to overestimate the role of multiplicative unitaries in the present theory of locally compact quantum groups. The concept introduced by Baaj and Skandalis [1] is present in purely theoretical considerations in axiomatic formulation of the theory [2, 3, 4]. It is also very useful, when one considers particular examples of quantum groups (cf. examples presented is Section 3). Usually having the Haar weight h on a quantum group G =(A, Δ) one uses the GNS-construction to define a Hilbert space H and an embedding A ⊂ B(H). Then the multiplicative unitary W is defined by taking the linear mapping: A ⊗ alg A ∋ a ⊗ b −→ Δ(a)(I ⊗ b) ∈ A ⊗ A and pushing it down to the level of H ⊗ H. This is so called Kac-Takesaki operator [4]. On the other hand in a number of cases, a multiplicative unitary is found independently and the general theory [1, 9, 5] is used to construct the corresponding locally compact quantum group. This was the way used in recent works [11, 12, 6] devoted to quantum deformations of the groups of affine transformations of R and C. In general the multiplicative unitary that we start with need not coincide with the Kac-Takesaki operator and the problem of existence of the Haar weight must be investigated. To pass from a multiplicative unitary W to the corresponding quantum group one has to assume that W have certain properties. For example it is sufficient to assume manageability [9] or at least modularity [5]. The Haar weight on the quantum ‘ax + b’ and ‘az + b’ groups was found by Van Daele some time ago [8]. To this end he had to use very particular properties of the groups. In autumn 2002, I played with the quantum ‘az + b’ group acting on the straight line girandole Γ (see formula (3.2) below). The aim was to show that a certain measure on Γ is relatively invariant. The problem reduced to a complicated formula relating the Fourier transform of a special function appearing in the theory with the holomorphic continuation of the function itself. To my surprise that was the same formula as the one used earlier to prove the modularity of the multiplicative unitary staying behind the quantum ‘az + b’ group. It was a strong indication that the (relative) invariance of some measure on a homogenous space is closely related to the modularity of the corresponding multiplicative unitary. Trying to explain this phenomenon when the homogenous space is the group itself (the case of proper homogenous spaces will be considered in a separate paper). I arrived to a simple formula describing a Haar weight on a quantum group coming from a modular multiplicative unitary. It is a special case of a formula found earier by Van Daele [8, 7]. The formula works if a certain technical assumption is satisfied. As a rule this is not the case for manageable multiplicative unitaries. Therefore the modularity introduced in [5] seems to be of real importance. We shortly describe the content of the paper. In section 1 we recall the basic concepts: mul- tiplicative unitaries, modularity and manageability and the passage from modular multiplicative unitaires to quantum groups. Next we explain what the Haar weight is. Then after a short dis- cussion of the trace and related weights we formulate our main theorem. It contains an explicit formula introducing a faithful lower semicontinuous weight h on a quantum group of the kind mentioned above. If h is locally finite then according to the theorem, h is the Haar weight. Section 1