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Lie Group Quantum Mathematics Seminar

Integrality of Kac-Moody groups

Lisa Carbone, Rutgers University

Location:  Hill 705
Date & time: Friday, 15 September 2023 at 12:10PM - 1:10PM

Let A be a symmetrizable generalized Cartan matrix with corresponding Kac-Moody algebra g=g(A) over Q. Let V be an integrable highest weight g-module and let V_Z be a Z-form of V. Let G=G(Q) be an associated minimal representation-theoretic Kac--Moody group and let G(Z) be its integral subgroup. By analogy with the finite dimensional case, the integrality question for G is to determine if G(Z)=Stab_G(V_Z), that is, to determine if G(Z) coincides with the subgroup of G that stabilizes the integral lattice V_Z. Integrality of semi-simple algebraic groups G over Q was established by Chevalley in the 1950’s as part of his work on associating an affine group scheme to G(Q) and V_Z. We discuss our approach to this question for Kac-Moody groups and we prove integrality for inversion subgroups U_w of G. Here, for w in W, U_w is the group generated by positive real root groups that are flipped to negative root groups by w^{-1}. There are various applications of this result, including integrality of subgroups of the unipotent subgroup U of G that are generated by commuting real root groups. This is joint work with Abid Ali, Dongwen Liu and Scott H. Murray.

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