Abstract. We give an explicit formula for invariant * -products on a wide class of coadjoint orbits. The answer is expressed in terms of the Shapovalov pairing for generalized Verma modules.

In [ABG] the derived category of the principal block in modules over the Lusztig quantum algebra at a root of unity is related to the derived category of equivariant coherent sheaves on the Springer resolution e N . In the present paper we deduce a similar relation between the derived category of the principal block for the small (reduced) quantum algebra u and the derived category of (non-equivariant) coherent sheaves on e N . As an application we get a geometric description of Hochschild cohomology (in particular, the center) of the regular block for u, and use it to give an explicit description of a certain subalgebra in the center (such a subalgebra was obtained previously by another method and under more restrictive assumptions in [La]). We also briefly explain the relation of our result to the geometric description [BK] of the derived category of modules over the De Concini -Kac quantum algebra.To the memory of Iosef Donin. Contents1. Quantum algebras 1.1. Basic notations and conventions 1.2. Quantum algebras at a root of unity 1.3. Quantum Lusztig-Frobenius map 2. Functor to a derived category of u-modules 2.1. The principal block 2.2. Springer resolution and U-modules 2.3. Springer resolution and u-modules 2.4. Connection to the Kac-De Concini algebra and a result of [BK] 3. Hochschild cohomology of the principal block of u 3.1. The result 3.2. The proof 3.3. An explicit subalgebra in the principal block of the center of the small quantum group.

We develop an elementary algebraic method to compute the center of the principal block of a small quantum group associated with a complex semisimple Lie algebra at a root of unity. The cases of sl 3 and sl 4 are computed explicitly. This allows us to formulate the conjecture that, as a bigraded vector space, the center of a regular block of the small quantum sl m at a root of unity is isomorphic to Haiman's diagonal coinvariant algebra for the symmetric group S m .

Using the quantum Fourier transform F [LM], we describe the block decomposition and multiplicative structure of a subalgebra Z + F ( Z) of the center of the small quantum group U f in q (g) at a root of unity. It contains the known subalgebra Z [BG], which is isomorphic to the algebra of characters of finite dimensional U f in q (g)-modules. We prove that the intersection Z ∩ F ( Z) coincides with the annihilator of the radical of Z. Applying representation-theoretical methods, we show that Z surjects onto the algebra of endomorphisms of certain indecomposable projective modules over U f in q (g). In particular this leads to the conclusion that the center of U f in q (g) coincides with Z + F ( Z) in the case g = sl 2 .