More Like this

1. A geometric formula for multiplicities of 𝐾-types of tempered representations

   https://doi.org/10.1090/tran/7857  

   Hochs, Peter ; Song, Yanli ; Yu, Shilin ( December 2019 , Transactions of the American Mathematical Society)

   Let G G be a connected, linear, real reductive Lie group with compact centre. Let K > G K>G be compact. Under a condition on K K , which holds in particular if K K is maximal compact, we give a geometric expression for the multiplicities of the K K -types of any tempered representation (in fact, any standard representation) π \pi of G G . This expression is in the spirit of Kirillov’s orbit method and the quantisation commutes with reduction principle. It is based on the geometric realisation of π | K \pi |_K obtained in an earlier paper. This expression was obtained for the discrete series by Paradan, and for tempered representations with regular parameters by Duflo and Vergne. We obtain consequences for the support of the multiplicity function, and a criterion for multiplicity-free restrictions that applies to general admissible representations. As examples, we show that admissible representations of SU ( p , 1 ) \textrm {SU}(p,1) , SO 0 ( p , 1 ) \textrm {SO}_0(p,1) , and SO 0 ( 2 , 2 ) \textrm {SO}_0(2,2) restrict multiplicity freely to maximal compact subgroups. 

   more » « less
2. Arakelov–Milnor inequalities and maximal variations of Hodge structure

   https://doi.org/10.1112/S0010437X23007157  

   Biquard, Olivier ; Collier, Brian ; García-Prada, Oscar ; Toledo, Domingo ( May 2023 , Compositio Mathematica)

   In this paper we study the $\mathbb {C}^*$ -fixed points in moduli spaces of Higgs bundles over a compact Riemann surface for a complex semisimple Lie group and its real forms. These fixed points are called Hodge bundles and correspond to complex variations of Hodge structure. We introduce a topological invariant for Hodge bundles that generalizes the Toledo invariant appearing for Hermitian Lie groups. An important result of this paper is a bound on this invariant which generalizes the Milnor–Wood inequality for a Hodge bundle in the Hermitian case, and is analogous to the Arakelov inequalities of classical variations of Hodge structure. When the generalized Toledo invariant is maximal, we establish rigidity results for the associated variations of Hodge structure which generalize known rigidity results for maximal Higgs bundles and their associated maximal representations in the Hermitian case. 

   more » « less
3. Affine cluster monomials are generalized minors

   https://doi.org/10.1112/S0010437X19007292  

   Rupel, Dylan ; Stella, Salvatore ; Williams, Harold ( June 2019 , Compositio Mathematica)

   We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with $\mathbf{g}$ -vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type $A_{1}^{(1)}$ , we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras. 

   more » « less
4. Representation Homology of Topological Spaces

   https://doi.org/10.1093/imrn/rnaa023  

   Berest, Yuri ; Ramadoss, Ajay C ; Yeung, Wai-Kit ( February 2020 , International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper, we introduce and study representation homology of topological spaces, which is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology parallel to the Loday–Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the two theories by proving that the representation homology of the suspension of a (pointed connected) space is isomorphic to its higher Hochschild homology. We also construct some natural maps and spectral sequences relating representation homology to other homology theories associated with spaces (such as Pontryagin algebras, ${{\mathbb{S}}}^1$-equivariant homology of the free loop space, and stable homology of automorphism groups of f.g. free groups). We compute representation homology explicitly (in terms of known invariants) in a number of interesting cases, including spheres, suspensions, complex projective spaces, Riemann surfaces, and some 3-dimensional manifolds, such as link complements in ${\mathbb{R}}^3$ and the lens spaces $ L(p,q) $. In the case of link complements, we identify the representation homology in terms of ordinary Hochschild homology, which gives a new algebraic invariant of links in ${\mathbb{R}}^3$. 

   more » « less
5. Period Relations and Special Values of Rankin-Selberg L-Functions

   https://doi.org/10.1007/978-3-319-59728-7_9  

   Harris, Michael ; Lin, Jie ( January 2017 , Representation Theory, Number Theory, and Invariant Theory)

   ThisisasurveyofrecentworkonvaluesofRankin-SelbergL-functions of pairs of cohomological automorphic representations that are critical in Deligne’s sense. The base field is assumed to be a CM field. Deligne’s conjecture is stated in the language of motives over Q, and express the critical values, up to rational factors, as determinants of certain periods of algebraic differentials on a projective algebraic variety over homology classes. The results that can be proved by automorphic methods express certain critical values as (twisted) period integrals of automorphic forms. Using Langlands functoriality between cohomological automorphic repre- sentations of unitary groups, which can be identified with the de Rham cohomology of Shimura varieties, and cohomological automorphic representations of GL.n/, the automorphic periods can be interpreted as motivic periods. We report on recent results of the two authors, of the first-named author with Grobner, and of Guerberoff. 

   more » « less