100 research outputs found
Finite difference quantum Toda lattice via equivariant K-theory
We construct the action of the quantum group U_v(sl_n) by the natural
correspondences in the equivariant localized -theory of the Laumon based
Quasiflags' moduli spaces. The resulting module is the universal Verma module.
We construct geometrically the Shapovalov scalar product and the Whittaker
vectors. It follows that a certain generating function of the characters of the
global sections of the structure sheaves of the Laumon moduli spaces satisfies
a -difference analogue of the quantum Toda lattice system, reproving the
main theorem of Givental-Lee. The similar constructions are performed for the
affine Lie agebra \hat{sl}_n.Comment: Some corrections are made in Sections 2,
Shifted quantum affine algebras: integral forms in type (with appendices by Alexander Tsymbaliuk and Alex Weekes)
We define an integral form of shifted quantum affine algebras of type and
construct Poincar\'e-Birkhoff-Witt-Drinfeld bases for them. When the shift is
trivial, our integral form coincides with the RTT integral form. We prove that
these integral forms are closed with respect to the coproduct and shift
homomorphisms. We prove that the homomorphism from our integral form to the
corresponding quantized -theoretic Coulomb branch of a quiver gauge theory
is always surjective. In one particular case we identify this Coulomb branch
with the extended quantum universal enveloping algebra of type . Finally, we
obtain the rational (homological) analogues of the above results (proved
earlier in arXiv:1611.06775, arXiv:1806.07519 via different techniques).Comment: v1: 65 pages; comments are welcome! v2: 67 pages; a dominance
condition is added in Section 2(vii), another definition is added in Appendix
A(viii), the injectivity of is added in Appendix B(ii).
v3: 70 pages; minor corrections, table of contents added, Section 3(vi)
updated and Remark 4.33 added following a new version of arXiv:1808.0953
Twisted geometric Satake equivalence
We generalize the classical Satake equivalence as follows. Let k be an
algebraically closed field, set O=k[[t]] and F=k((t)). For an almost simple
algebraic group G we classify central extensions of G(F) by the multiplicative
group. Any such extension E splits canonically over G(O). Consider the category
of G(O)-biinvariant perverse sheaves on E with a given Gm-monodromy . We show
that this is a tensor category, which is tensor equivalent to the category of
representations of a reductive group. We compute the root datum of this group.Comment: 22 pages, a reference to Lusztig is added. Final version to appear in
J. of the Institute of Math. of Jussie
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