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Register a new userA geometric version of BGP reflection functors
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Stefan Wolf
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Quiver Grassmannians and quiver flags are natural generalisations of usual Grassmannians and flags. They arise in the study of quiver representations and Hall algebras. In general, they are projective varieties which are neither smooth nor irreducible. We use a scheme theoretic approach to calculate their tangent space and to obtain a dimension estimate similar to one of Reineke. Using this we can show that if there is a generic representation, then these varieties are smooth and irreducible. If we additionally have a counting polynomial we deduce that their Euler characteristic is positive and that the counting polynomial evaluated at zero yields one. After having done so, we introduce a geometric version of BGP reflection functors which allows us to deduce an even stronger result about the constant coefficient of the counting polynomial. We use this to obtain an isomorphism between the Hall algebra at q=0 and Reinekes generic extension monoid in the Dynkin case.
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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 algebra, one can apply the BGP-reflection functors to the double Ringel-Hall algebra to obtain Lusztigs symmetries on $mathbf{U}$ and their important properties, for instance, the braid relations. In this paper, we define a modified form $dot{mathcal{H}}$ of the Ringel-Hall algebra and realize the Lusztigs symmetries on $dot{mathbf{U}}$ by applying the BGP-reflection functors to $dot{mathcal{H}}$.
We give another proof, using tools from Geometric Invariant Theory, of a result due to S. Sam and A. Snowden in 2014, concerning the stability of Kro-necker coefficients. This result states that some sequences of Kronecker coefficients eventually stabilise, and our method gives a nice geometric bound from which the stabilisation occurs. We perform the explicit computation of such a bound on two examples, one being the classical case of Murnaghans stability. Moreover, we see that our techniques apply to other coefficients arising in Representation Theory: namely to some plethysm coefficients and in the case of the tensor product of representations of the hyperoctahedral group.
Let G be a simple simply-connected group over an algebraically closed field k, X be a smooth connected projective curve over k. In this paper we develop the theory of geometric Eisenstein series on the moduli stack Bun_G of G-torsors on X in the setting of the quantum geometric Langlands program (for etale l-adic sheaves) in analogy with [3]. We calculate the intersection cohomology sheaf on the version of Drinfeld compactification in our twisted setting. In the case G=SL_2 we derive some results about the Fourier coefficients of our Eisenstein series. In the case of G=SL_2 and X=P^1 we also construct the corresponding theta-sheaves and prove their Hecke property.
These myh lectures at the Park City conference in 1998.
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 on a semi-simple group over a local non-Archimedian field, and Grothendieck group of equivariant coherent sheaves on Steinberg variety of the Langlands dual group; this isomorphism due to Kazhdan--Lusztig and Ginzburg is a key step in the proof of tamely ramified local Langlands conjectures. The paper is a continuation of an earlier joint work with S. Arkhipov, it relies on technical material developed in a paper with Z. Yun.
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