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In this paper, we give geometric realizations of Lusztigs symmetries. We also give projective resolutions of a kind of standard modules. By using the geometric realizations and the projective resolutions, we obtain the categorification of the formulas of Lusztigs symmetries.
The geometric realizations of Lusztigs symmetries of symmetrizable quantum groups are given in this paper. This construction is a generalization of that in [19].
In this paper, we shall study the structure of the Grothendieck group of the category consisting of Lusztigs perverse sheaves and give a decomposition theorem of it. By using this decomposition theorem and the geometric realizations of Lusztigs symme
Let $mathbf{U}$ be the quantized enveloping algebra and $dot{mathbf{U}}$ its modified form. Lusztig gives some symmetries on $mathbf{U}$ and $dot{mathbf{U}}$. Since the realization of $mathbf{U}$ by the reduced Drinfeld double of the Ringel-Hall alge
Our investigation in the present paper is based on three important results. (1) In [12], Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the qua
The article is a contribution to the local theory of geometric Langlands correspondence. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra, thought of as an algebra of Iwahori bi-invariant functions