2021/01/06 by Yury A. Neretin, Neretin, Yury A.
Mathematics · #05E05 #20C08 #20H30 #43A10 #54H11 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.GR #math.RA #math.RT #msc:05E05 #msc:20C08 #msc:20H30 #msc:43A10 #msc:54H11
paper · pdf · doi:10.48550/arxiv.2101.02133
23p
arxiv created 2021/01/06 · openalex publication_date 2021/01/06 · arxiv updated 2021/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The infinite-dimensional Iwahori--Hecke algebras H_∞(q) are direct limits of the usual finite-dimensional Iwahori--Hecke algebras. They arise in a natural way as convolution algebras of bi-invariant functions on groups GLB(\mathbbFq) of infinite-dimensional matrices over finite-fields having only finite number of non-zero matrix elements under the diagonal. In 1988 Vershik and Kerov classified all indecomposable positive traces on H_∞(q). Any such trace generates a representation of the double H_∞(q)⊗ H_∞(q) and of the double GLB(\mathbbFq)× GLB(\mathbbFq). We present constructions of such representations; the traces are some distinguished matrix elements. We also obtain some (simple) general statements on relations between unitary representations of groups and representations of convolution algebras of measures bi-invariant with respect to compact subgroups.