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Register a new userA nonabelian Fourier transform for tempered unipotent representations
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We define an involution on the space of compact tempered unipotent representations of inner twists of a split simple $p$-adic group $G$ and investigate its behaviour with respect to restrictions to reductive quotients of maximal compact open subgroups. In particular, we formulate a precise conjecture about the relation with a version of Lusztigs nonabelian Fourier transform on the space of unipotent representations of the (possibly disconnected) reductive quotients of maximal compact subgroups. We give evidence of the conjecture, including proofs for $mathsf{SL}_n$ and $mathsf{PGL}_n$.
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Let $G$ be a real classical group of type $B$, $C$, $D$ (including the real metaplectic group). We consider a nilpotent adjoint orbit $check{mathcal O}$ of $check G$, the Langlands dual of $G$ (or the metaplectic dual of $G$ when $G$ is a real metaplectic group). We classify all special unipotent representations of $G$ attached to $check{mathcal O}$, in the sense of Barbasch and Vogan. When $check{mathcal O}$ is of good parity, we construct all such representations of $G$ via the method of theta lifting. As a consequence of the construction and the classification, we conclude that all special unipotent representations of $G$ are unitarizable, as predicted by the Arthur-Barbasch-Vogan conjecture. We also determine precise structure of the associated cycles of special unipotent representations of $G$.
In [Frobenius1896] it was shown that many important properties of a finite group could be examined using formulas involving the character ratios of group elements, i.e., the trace of the element acting in a given irreducible representation, divided by the dimension of the representation. In [Gurevich-Howe15] and [Gurevich-Howe17], the current authors introduced the notion of rank of an irreducible representation of a finite classical group. One of the motivations for studying rank was to clarify the nature of character ratios for certain elements in these groups. In fact in the above cited papers, two notions of rank were given. The first is the Fourier theoretic based notion of U-rank of a representation, which comes up when one looks at its restrictions to certain abelian unipotent subgroups. The second is the more algebraic based notion of tensor rank which comes up naturally when one attempts to equip the representation ring of the group with a grading that reflects the central role played by the few smallest possible representations of the group. In [Gurevich-Howe17] we conjectured that the two notions of rank mentioned just above agree on a suitable collection called low rank representations. In this note we review the development of the theory of rank for the case of the general linear group GL_n over a finite field F_q, and give a proof of the agreement conjecture that holds true for sufficiently large q. Our proof is Fourier theoretic in nature, and uses a certain curious positivity property of the Fourier transform of the set of matrices of low enough fixed rank in the vector space of matrices of size m x n over F_q. In order to make the story we are trying to tell clear, we choose in this note to follow a particular example that shows how one might apply the theory of rank to certain counting problems.
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.
We show that Fourier coefficients of automorphic forms attached to minimal or next-to-minimal automorphic representations of ${mathrm{SL}}_n(mathbb{A})$ are completely determined by certain highly degenerate Whittaker coefficients. We give an explicit formula for the Fourier expansion, analogously to the Piatetski-Shapiro-Shalika formula. In addition, we derive expressions for Fourier coefficients associated to all maximal parabolic subgroups. These results have potential applications for scattering amplitudes in string theory.
With the aid of the exponentiation functor and Fourier transform we introduce a class of modules $T(g,V,S)$ of $mathfrak{sl} (n+1)$ of mixed tensor type. By varying the polynomial $g$, the $mathfrak{gl}(n)$-module $V$, and the set $S$, we obtain important classes of weight modules over the Cartan subalgebra $mathfrak h$ of $mathfrak{sl} (n+1)$, and modules that are free over $mathfrak h$. Furthermore, these modules are obtained through explicit presentation of the elements of $mathfrak{sl} (n+1)$ in terms of differential operators and lead to new tensor coherent families of $mathfrak{sl} (n+1)$. An isomorphism theorem and simplicity criterion for $T(g,V,S)$ is provided.
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