lattices-in-lie-groups

Hasse diagrams of G/P_1 and G/P_2

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in the Paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.30.5052&rep=rep1&type=pdf at the end, we can see Hasse diagrams for several projective, homogeneous G-varieties for G being a exceptional linear algebraic group. Note that D4/P1 is isomorphic to a six dimensional quadric, that i will denote as Q6. In an unfinished book by Gille, Petrov,N. Semenov and Zainoulline, which can be found … Read more

Milnor’s model of EGEG and Kac-Moody groups

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I am working with non-compact Kac-Moody groups K. We can use Milnor’s join model for EK=lim→K∗n, where K∗n is the iterated join (see page 20 of this PDF). Let KJ be a parabolic subgroup of K. I would like to use the system {K∗n} as a model for EKJ and compare it to the system … Read more

classification of homogenous complex manifolds

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Suppose X is a complex manifold (doesn’t assume it’s Kahler), and it’s holomorhpic automorphism group is transitive. My question is that is there any classification of those manifolds ? Answer AttributionSource : Link , Question Author : user42804 , Answer Author : Community

When can a locally compact group be approximated by discrete subgroups?

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This question is about partitioning a (locally) compact group into cells by using discrete subgroups. Let G be a locally compact group. (I am really most interested in the case where G is a metrizable group.) Say that G is “approximable by discrete subgroups” if there is a sequence of countable/finite locally compact subgroups G1⊆G2⊆G3⊆…⊆G … Read more

Compactly supported distributions as a projective G-module

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For a Lie group G and a locally convex space V let E(G,V) be the locally convex space of smooth functions from G to V, and accordingly E′c(G,V) the space of compactly supported distributions. A G-module V is called differentiable if v→(g→gv) defines a continuous map from V to E(G,V) for all v∈V. In particular … Read more

Sobolev spaces defined on non-compact Lie groups

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In this post, a question was raised to discuss the generalization of Sobolev spaces on locally compact Lie groups. Now my question is whether there exists a generalization of Sobolev spaces and associated theories of the Laplace operator on noncompact Lie groups. My expectation is that the theory should be reduced to coincide with that … Read more

The homotopy type of the mapping space MapBρ(BS1,BG)Map_{B\rho}(BS^1,BG)? for GG a compact Lie group

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Given a homomorphism ρ:S1→G with G a compact Lie group there is an induced map of classifying spaces Bρ:BS1→BG. What is known about the homotopy type of the mapping space MapBρ(BS1,BG)? Integrally the space will be a mess of phantom maps. On the other hand, after rationalisation, BGQ will be a product of even dimensional … Read more

Harmonicity on semisimple groups

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I asked this on Math.SE and got no answer, so I’ll try my luck here. Let G be a semisimple real Lie group, U(g) its universal enveloping algebra, let Ω be the Casimir element in U(g) and let f be a smooth (or analytic) real-valued function on G. We then have the following notions 1) … Read more

Points of failure in definition of X- and A-moduli spaces for arbitrary G

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In their work [0] on defining notions of higher Teichmüller space for local systems on surfaces, Fock and Goncharov require split reductive Lie groups, and sometimes also require simple-connectedness. What properties of these groups are required for the definition—in particular, what properties of Borel subgroups do Fock and Goncharov assume that, in general, do not … Read more

A few questions about E_7E_7 and its symmetric spaces

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My question about E_6 survived, so I post next episode. From the Yokota book I found out that there is -1 in E_7 Lie group. This book defines Lie group E_7 using 56-dimensional Freudenthal vector space over complex numbers. I prefer to use 28-dimensional quaternion space for representation of E_7. See also Wilson papers with … Read more