Indira Chatterji
We apply V. Lafforgue's techniques to establish the rapid decay property for cocompact lattices in a finite product of rank one Lie groups with Lie groups whose restricted root system is of type A2.
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We initiate an investigation of lattices in a new class of locally compact groups, so called locally pro-$p$-complete Kac-Moody groups. We discover that in rank 2 their cocompact lattices are particularly well-behaved: under mild assumptions, a cocompact lattice in this completion contains no elements of order $p$. This statement is still an open question for the Caprace-Rémy-Ronan completion. Using this, modulo results of Capdeboscq and Thomas, we classify edge-transitive cocompact lattices and describe a cocompact lattice of minimal covolume.
We derive a formula for the regularized trace of operators with compact spectrum which act on the space of square integrable functions on the quotient of a semisimple Liegroup of real rank one by a convex-cocompact subgroup. The sum of normalized orbital integrals associated to the hyperbolic conjugacy classes of this subgroup is an invariant distribution on the group, and we state precise conjecture about its Fouriertransform. We apply our trace formula to resolvent kernels. This yields functional equations of Selberg zeta functions and information about their singularities. The paper leaves …
We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding $\mathbb{N}\trianglelefteq\mathbb{E}$ of locally compact groups and a twisted action $(α,τ)$ thereof on a (post)liminal $C^*$-algebra $A$ the twisted crossed product $A\rtimes_{α,τ}\mathbb{E}$ is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup $\mathbb{N}\trianglelefteq \mathbb{E}$ is type-I as soon as $\mathbb{E}$ is. This happens for instance if $\mathbb{N}$ i…
For any solvable Lie group whose exponential map $\exp_G\colon{\mathfrak g}\to G$ is bijective, we prove that the real rank of $C^*(G)$ is equal to $\dim({\mathfrak g}/[{\mathfrak g},{\mathfrak g}])$. We also indicate a proof of a similar formula for the stable rank of $C^*(G)$, as well as some estimates on the ideal generated by the projections in $C^*(G)$.
Let $G$ be a Lie-group and $\Ga\subset G$ a cocompact lattice. For a finite-dimensional, not necessarily unitary representation $\om$ of $\Ga$ we show that the $G$-representation on $L^2(\Ga\bs G,\om)$ admits a complete filtration with irreducible quotients. As a consequence, we show the trace formula for non-unitary twists and arbitrary locally compact groups.
Extensions of finite irreducible modules over rank two Lie conformal algebra
本文完整分类了秩二李共形代数上有限不可约共形模的扩张，解决了该代数表示论中的模扩张分类问题。
We give a complete classification (up to isomorphism) of Lie conformal algebras which are free of rank two as $\C[\partial]$-modules, and determine their automorphism groups.
In the present paper, we prove that any finite non-trivial irreducible module over a rank two Lie conformal algebra $\mathcal{H}$ is of rank one. We also describe the actions of $\mathcal{H}$ on its finite irreducible modules explicitly. Moreover, we show that all finite non-trivial irreducible modules of finite Lie conformal algebras whose semisimple quotient is the Virasoro Lie conformal algebra are of rank one.
We determine all spherical algebraic subalgebras in any simple Lie algebra of real rank 1.
We study the truncation functors and show the existence of projective cover of each irreducible module in parabolic BGG category $\mathcal O$ over infinite rank Lie algebra of types $\mathfrak{a,b,c,d}$. Moreover, $\mathcal O$ is a Koszul category. As a consequence, the corresponding parabolic BGG category $\overline{\mathcal O}$ over infinite rank Lie superalgebra of types $\mathfrak{a,b,c,d}$ through the super duality is also a Koszul category.
We obtain the Plancherel theorem for the quotient of a simple Lie group of real rank one by a convex-cocompact discrete subgroup and its consequences for the spectrum of locally invariant differential operators on bundles over Kleinian manifolds. We develop a geometric version of scattering theory. The paper is an update of dg-ga/9609011, which is completely rewritten from the point of view of representation theory. In particular, we are now able to show the meromorphy of Eisenstein series for all parameters.
Gelfand-Tsetlin modules for Lie algebras of rank $2$
本文显式构造秩 $2$ 李代数上的单模族，使交换子代数对角作用且谱为单谱，从而将 $A$ 型泛型 Gelfand-Tsetlin 模推广到其他秩 $2$ 情形。
There exist principal $\mathfrak{sl}_2$ subalgebras for hyperbolic Kac-Moody Lie algebras. In the case of rank 2 symmetric hyperbolic Kac-Moody Lie algebras, certain $\mathfrak{sl}_2$ subalgebras are constructed. These subalgebras are generalizations of principal $\mathfrak{sl}_2$ subalgebras. We show that the rank 2 symmetric hyperbolic Kac-Moody Lie algebras themselves are irreducibly decomposed under the action of this $\mathfrak{sl}_2$ subalgebras. Furthermore, we classify irreducible components of the decomposition. In particular, we obtain multiplicities of unitary principal series and c…
In analogy to the theory of nilpotent orbit in finite-dimensional semisimple Lie algebras, it is known that the principal $\mathfrak{sl}_2$ subalgebras can be constructed in hyperbolic Kac-Moody Lie algebras. We obtained a series of $\mathfrak{sl}_2$ subalgebras in rank 2 symmetric hyperbolic Kac-Moody Lie algebras by extending the aforementioned construction. We present this result and also discuss $\mathfrak{sl}_2$ modules obtained by the action of the $\mathfrak{sl}_2$ subalgebras on the original Lie algebras.
In the high rank limit, the fraction of non-zero character table entries of finite simple groups of Lie type goes to zero.
In this short note, we study the rank of a restricted Lie algebra $(\mathfrak{g},[p])$ and give some applications, which concerns the dimensions of non-trivial irreducible modules. We also compute the rank of the restricted contact algebra $K(n)$.
Localised analytic torsion and relative analytic torsion for non compact Lie groups of type I
本文针对I型非紧李群定义表示局部化解析挠率，并据此定义群与离散余紧子群配对的相对解析挠率，证明其等于覆盖空间的Lott $L^2$解析挠率，且以Abel群和Heisenberg群为例说明。
For any prime $p$ and $S$ a $p$-group isomorphic to a Sylow $p$-subgroup of a rank $2$ simple group of Lie type in characteristic $p$, we determine all saturated fusion systems supported on $S$ up to isomorphism.
We classify the quasi-finite irreducible highest weight modules over the infinite rank Lie superalgebras $\hgltwo$, $\hC$ and $\hD$, and determine the necessary and sufficient conditions for quasi-finite irreducible highest weight modules to be unitarizable with respect to natural $\ast$-structures of the Lie superalgebras. The unitarizable irreducible modules are constructed in terms of Fock spaces of free quantum fields, and explicit formulae for their formal characters are also obtained by investigating Howe dualities between the infinite rank Lie superalgebras and classical Lie groups.
A procedure is described that makes use of the generating function of characters to obtain a new generating function $H$ giving the multiplicities of each weight in all the representations of a simple Lie algebra. The way to extract from $H$ explicit multiplicity formulas for particular weights is explained and the results corresponding to rank two simple Lie algebras shown.
Classification of simple differential Lie and Jordan (super)coalgebras of finite rank
本文分类了有限秩单微分李与若尔当（超）余代数，并显式描述了 n=2,3,4 超共形李代数及例外李共形超代数 CK$_6$ 的 OPE 所对应的李超余代数。
Classical invariants for representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of classical invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on the moduli stack of parabolic bundles on a smooth curve. Rank-level duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra. In this paper we prove a rank-level duality for type so(2r…
A fixed point theorem for the action of linear higher rank algebraic groups over local fields on symmetric spaces of infinite dimension and finite rank
论文研究了局部域上线性代数群在无限维有限秩对称空间上的连续等距作用，证明了存在不动点，并探讨了SL_3(F)条件下的扩展情况。
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