{"title":"The Gelfand–Graev representation of classical groups in terms of Hecke algebras","authors":"Petar Bakić, Gordan Savin","doi":"10.4153/S0008414X2200030X","DOIUrl":"https://doi.org/10.4153/S0008414X2200030X","url":null,"abstract":"Abstract Let G be a p-adic classical group. The representations in a given Bernstein component can be viewed as modules for the corresponding Hecke algebra—the endomorphism algebra of a pro-generator of the given component. Using Heiermann’s construction of these algebras, we describe the Bernstein components of the Gelfand–Graev representation for \u0000$G=mathrm {SO}(2n+1)$\u0000 , \u0000$mathrm {Sp}(2n)$\u0000 , and \u0000$mathrm {O}(2n)$\u0000 .","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"32 1","pages":"1343 - 1368"},"PeriodicalIF":0.7,"publicationDate":"2022-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89553201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Classification of generalized Einstein metrics on three-dimensional Lie groups","authors":"V. Cort'es, David Krusche","doi":"10.4153/S0008414X23000056","DOIUrl":"https://doi.org/10.4153/S0008414X23000056","url":null,"abstract":"Abstract We develop the theory of left-invariant generalized pseudo-Riemannian metrics on Lie groups. Such a metric accompanied by a choice of left-invariant divergence operator gives rise to a Ricci curvature tensor, and we study the corresponding Einstein equation. We compute the Ricci tensor in terms of the tensors (on the sum of the Lie algebra and its dual) encoding the Courant algebroid structure, the generalized metric, and the divergence operator. The resulting expression is polynomial and homogeneous of degree 2 in the coefficients of the Dorfman bracket and the divergence operator with respect to a left-invariant orthonormal basis for the generalized metric. We determine all generalized Einstein metrics on three-dimensional Lie groups.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"51 1","pages":"2038 - 2095"},"PeriodicalIF":0.7,"publicationDate":"2022-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76050925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Multilinear transference of Fourier and Schur multipliers acting on noncommutative \u0000$L_p$\u0000 -spaces","authors":"M. Caspers, Amudhan Krishnaswamy-Usha, G. Vos","doi":"10.4153/S0008414X2200058X","DOIUrl":"https://doi.org/10.4153/S0008414X2200058X","url":null,"abstract":"Abstract Let G be a locally compact unimodular group, and let \u0000$phi $\u0000 be some function of n variables on G. To such a \u0000$phi $\u0000 , one can associate a multilinear Fourier multiplier, which acts on some n-fold product of the noncommutative \u0000$L_p$\u0000 -spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an n-fold product of Schatten classes \u0000$S_p(L_2(G))$\u0000 . We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called “multiplicatively bounded \u0000$(p_1,ldots ,p_n)$\u0000 -norm” of a multilinear Schur multiplier is bounded above by the corresponding multiplicatively bounded norm of the Fourier multiplier, with equality whenever the group is amenable. Furthermore, we prove that the bilinear Hilbert transform is not bounded as a vector-valued map \u0000$L_{p_1}(mathbb {R}, S_{p_1}) times L_{p_2}(mathbb {R}, S_{p_2}) rightarrow L_{1}(mathbb {R}, S_{1})$\u0000 , whenever \u0000$p_1$\u0000 and \u0000$p_2$\u0000 are such that \u0000$frac {1}{p_1} + frac {1}{p_2} = 1$\u0000 . A similar result holds for certain Calderón–Zygmund-type operators. This is in contrast to the nonvector-valued Euclidean case.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"224 1","pages":"1986 - 2006"},"PeriodicalIF":0.7,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80083888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}