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Register a new userRepresentations of Deligne-Mostow lattices into PGL(3, C)
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E Falbel
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We classify representations of a class of Deligne-Mostow lattices into PGL(3, C). In particular, we show local rigidity for the representations (of Deligne-Mostow lattices with 3-fold symmetry and of type one) where the generators we chose are in the same conjugacy class as the generators of Deligne-Mostow lattices. We use formal computations in SAGE to obtain the results. The code files are available on GitHub ([FPUP21]).
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As for the theory of maximal representations, we introduce the volume of a Zimmers cocycle $Gamma times X rightarrow mbox{PO}^circ(n, 1)$, where $Gamma$ is a torsion-free (non-)uniform lattice in $mbox{PO}^circ(n, 1)$, with $n geq 3$, and $X$ is a suitable standard Borel probability $Gamma$-space. Our numerical invariant extends the volume of representations for (non-)uniform lattices to measurable cocycles and in the uniform setting it agrees with the generalized version of the Euler number of self-couplings. We prove that our volume of cocycles satisfies a Milnor-Wood type inequality in terms of the volume of the manifold $Gamma backslash mathbb{H}^n$. This invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a measurable map $X rightarrow mbox{PO}(n, 1)$ with essentially constant sign. As a by-product of our rigidity result for the volume of cocycles, we give a new proof of the mapping degree theorem. This allows us to provide a complete characterization of maps homotopic to local isometries between closed hyperbolic manifolds in terms of maximal cocycles. In dimension $n = 2$, we introduce the notion of Euler number of measurable cocycles associated to closed surface groups. It extends the classic Euler number of representations and it agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. We show a Milnor-Wood type inequality whose upper bound is given by the modulus of the Euler characteristic. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization.
We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne-Lusztig (or Alvis-Curtis) duality for $p$-adic groups and the homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for $G=GL(n)$ by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct we obtain a description of the Serre functor for representations of a p-adic group.
This is Part IV of a thematic series currently consisting of a monograph and four essays. This essay examines the form of induced representations of locally p-adic Lie groups G which is appropriate for the abelian category of ${mathcal M}_{c}(G)$-admissible representations. In my non-expert manner, I prove the analogue of Jacquets Theorem in this category. The final section consists of observations and questions related to this and other concepts introduced in the course of this series.
The Gell-Mann grading, one of the four gradings of sl(3,C) that cannot be further refined, is considered as the initial grading for the graded contraction procedure. Using the symmetries of the Gell-Mann grading, the system of contraction equations is reduced and solved. Each non-trivial solution of this system determines a Lie algebra which is not isomorphic to the original algebra sl(3,C). The resulting 53 contracted algebras are divided into two classes - the first is represented by the algebras which are also continuous Inonu-Wigner contractions, the second is formed by the discrete graded contractions.
Rapoport and Kottwitz defined the affine Deligne-Lusztig varieties $X_{tilde{w}}^P(bsigma)$ of a quasisplit connected reductive group $G$ over $F = mathbb{F}_q((t))$ for a parahoric subgroup $P$. They asked which pairs $(b, tilde{w})$ give non-empty varieties, and in these cases what dimensions do these varieties have. This paper answers these questions for $P=I$ an Iwahori subgroup, in the cases $b=1$, $G=SL_2$, $SL_3$, $Sp_4$. This information is used to get a formula for the dimensions of the $X_{tilde{w}}^K(sigma)$ (all shown to be non-empty by Rapoport and Kottwitz) for the above $G$ that supports a general conjecture of Rapoport. Here $K$ is a special maximal compact subgroup.
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