9,488 research outputs found
Property (T) and rigidity for actions on Banach spaces
We study property (T) and the fixed point property for actions on and
other Banach spaces. We show that property (T) holds when is replaced by
(and even a subspace/quotient of ), and that in fact it is
independent of . We show that the fixed point property for
follows from property (T) when 1
. For simple Lie groups and their lattices, we prove that the fixed point property for holds for any if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces.Comment: Many minor improvement
Affine Buildings and Tropical Convexity
The notion of convexity in tropical geometry is closely related to notions of
convexity in the theory of affine buildings. We explore this relationship from
a combinatorial and computational perspective. Our results include a convex
hull algorithm for the Bruhat--Tits building of SL and techniques for
computing with apartments and membranes. While the original inspiration was the
work of Dress and Terhalle in phylogenetics, and of Faltings, Kapranov, Keel
and Tevelev in algebraic geometry, our tropical algorithms will also be
applicable to problems in other fields of mathematics.Comment: 22 pages, 4 figure
On p-adic lattices and Grassmannians
It is well-known that the coset spaces G(k((z)))/G(k[[z]]), for a reductive
group G over a field k, carry the geometric structure of an inductive limit of
projective k-schemes. This k-ind-scheme is known as the affine Grassmannian for
G. From the point of view of number theory it would be interesting to obtain an
analogous geometric interpretation of quotients of the form
G(W(k)[1/p])/G(W(k)), where p is a rational prime, W denotes the ring scheme of
p-typical Witt vectors, k is a perfect field of characteristic p and G is a
reductive group scheme over W(k). The present paper is an attempt to describe
which constructions carry over from the function field case to the p-adic case,
more precisely to the situation of the p-adic affine Grassmannian for the
special linear group G=SL_n. We start with a description of the R-valued points
of the p-adic affine Grassmannian for SL_n in terms of lattices over W(R),
where R is a perfect k-algebra. In order to obtain a link with geometry we
further construct projective k-subvarieties of the multigraded Hilbert scheme
which map equivariantly to the p-adic affine Grassmannian. The images of these
morphisms play the role of Schubert varieties in the p-adic setting. Further,
for any reduced k-algebra R these morphisms induce bijective maps between the
sets of R-valued points of the respective open orbits in the multigraded
Hilbert scheme and the corresponding Schubert cells of the p-adic affine
Grassmannian for SL_n.Comment: 36 pages. This is a thorough revision, in the form accepted by Math.
Zeitschrift, of the previously published preprint "On p-adic loop groups and
Grassmannians
Brane Realizations of Quantum Hall Solitons and Kac-Moody Lie Algebras
Using quiver gauge theories in (1+2)-dimensions, we give brane realizations
of a class of Quantum Hall Solitons (QHS) embedded in Type IIA superstring on
the ALE spaces with exotic singularities. These systems are obtained by
considering two sets of wrapped D4-branes on 2-spheres. The space-time on which
the QHS live is identified with the world-volume of D4-branes wrapped on a
collection of intersecting 2-spheres arranged as extended Dynkin diagrams of
Kac-Moody Lie algebras. The magnetic source is given by an extra orthogonal
D4-brane wrapping a generic 2-cycle in the ALE spaces. It is shown as well that
data on the representations of Kac-Moody Lie algebras fix the filling factor of
the QHS. In case of finite Dynkin diagrams, we recover results on QHS with
integer and fractional filling factors known in the literature. In case of
hyperbolic bilayer models, we obtain amongst others filling factors describing
holes in the graphene.Comment: Lqtex; 15 page
Demazure resolutions as varieties of lattices with infinitesimal structure
Let k be a field of positive characteristic. We construct, for each dominant
coweight \lambda of the standard maximal torus in the special linear group, a
closed subvariety D(\lambda) of the multigraded Hilbert scheme of an affine
space over k, such that the k-valued points of D(\lambda) can be interpreted as
lattices in k((z))^n endowed with infinitesimal structure. Moreover, for any
\lambda we construct a universal homeomorphism from D(\lambda) to a Demazure
resolution of the Schubert variety associated with \lambda in the affine
Grassmannian. Lattices in D(\lambda) have non-trivial infinitesimal structure
if and only if they lie over the boundary of the big cell.Comment: 24 pages, added the missing bibliograph
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