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Register a new userRepresentations of the Infinite-Dimensional $p$-Adic Affine Group
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We introduce an infinite-dimensional $p$-adic affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by the fact that the group does not act on the phase space. However it is possible to define its action on some classes of functions.
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We begin this paper by reviewing the Langlands correspondence for unipotent representations of the exceptional group of type $G_2$ over a $p$-adic field $F$ and present it in an explicit form. Then we compute all ABV-packets, as defined in [CFM+21] following ideas from Vogans 1993 paper The local Langlands Conjecture, and prove that these packets satisfy properties derived from the expectation that they are generalized A-packets. We attach distributions to ABV-packets for $G_2$ and its endoscopic groups and study a geometric endoscopic transfer of these distributions. This paper builds on earlier work by the same authors.
The general construction of frames of p-adic wavelets is described. We consider the orbit of a mean zero generic locally constant function with compact support (mean zero test function) with respect to the action of the p-adic affine group and show that this orbit is a uniform tight frame. We discuss relation of this result to the multiresolution wavelet analysis.
Motivated by physical and topological applications, we study representations of the group $mathcal{LB}_3$ of motions of $3$ unlinked oriented circles in $mathbb{R}^3$. Our point of view is to regard the three strand braid group $mathcal{B}_3$ as a subgroup of $mathcal{LB}_3$ and study the problem of extending $mathcal{B}_3$ representations. We introduce the notion of a emph{standard extension} and characterize $mathcal{B}_3$ representations admiting such an extension. In particular we show, using a classification result of Tuba and Wenzl, that every irreducible $mathcal{B}_3$ representation of dimension at most $5$ has a (standard) extension. We show that this result is sharp by exhibiting an irreducible $6$-dimensional $mathcal{B}_3$ representation that has no extensions (standard or otherwise). We obtain complete classifications of (1) irreducible $2$-dimensional $mathcal{LB}_3$ representations (2) extensions of irreducible $3$-dimensional $mathcal{B}_3$ representations and (3) irreducible $mathcal{LB}_3$ representations whose restriction to $mathcal{B}_3$ has abelian image.
We prove the twisted Whittaker category on the affine flag variety and the category of representations of the mixed quantum group are equivalent.
Inspired by the work [Ra1], we directly give a complete classification of irreducible calibrated representations of affine Yokonuma-Hecke algebras $widehat{Y}_{r,n}(q)$ over $mathbb{C},$ which are indexed by $r$-tuples of placed skew shapes. We then develop several applications of this result. In the appendix, inspired by [Ru], we classify and construct irreducible completely splittable representations of degenerate affine Yokonuma-Hecke algebras $D_{r,n}$ and the wreath product $(mathbb{Z}/rmathbb{Z})wr mathfrak{S}_{n}$ over an algebraically closed field of characteristic $p> 0$ such that $p$ does not divide $r$.
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