On the Elliptic Nonabelian Fourier Transform for Unipotent Representations of p-Adic Groups

Abstract

Abstract. In this paper, we consider the relation between two nonabelian Fourier transforms. The first one is defined in terms of the Langlands-Kazhdan-Lusztig parameters for unipotent elliptic representations of a split p-adic group and the second is defined in terms of the pseudocoefficients of these representations and Lusztig's nonabelian Fourier transform for characters of finite groups of Lie type. We exemplify this relation in the case of the p-adic group of type G 2 . In this paper, we consider the relation between two nonabelian Fourier transforms: the first defined in terms of the Langlands-Kazhdan-Lusztig parameters for unipotent elliptic representations of a split p-adic group and the second, defined in terms of the pseudocoefficients of these representations and Lusztig's nonabelian Fourier transform for characters of finite groups of Lie type. In this introduction, we give a brief outline of the ideas involved, leaving the precise definitions for the main body of the paper.Let G be a semisimple p-adic group. Lusztig [Lu1] defined the category C u (G) of unipotent Grepresentations. An informal characterization of C u (G) is that it is the smallest full subcategory of smooth G-representations such that: (a) it contains the irreducible representations with Iwahori-fixed vectors [Bo], and (b) it is closed under partition into L-packets. Let R u (G) be the complexification of the Grothendieck group of admissible representations in C u (G). Let H(G) be the Hecke algebra of G with respect to a Haar measure on G and let H(G) be the cocenter of H(G).If one is interested in the study of characters of admissible representations, the basic case is that of elliptic tempered representations ([Ar, BDK]). Let R u (G) ell be the elliptic representation space, a certain subspace of R u (G) isomorphic to the quotient of R u (G) by the span of all properly parabolically induced characters. To every elliptic character π ∈ R u (G) ell , one attaches a pseudocoefficient f π ∈ H(G) ([Ar, Ka2, SS]). The functions f π play an important role in the character formulas for elliptic representations: for example, if π is an irreducible square integrable representation, then f π (1) is the formal degree of π, e.g., [SS, Proposition III.4.4]. Let H(G) ell u denote the subspace of H(G) spanned by the pseudocoefficients f π for π ∈ R u (G) ell . Thus, we have a map, in fact, an isomorphism, R u (G) ell → H(G)

Toward the endoscopic classification of unipotent representations of $p$-adic $G_2$

“…Elliptic unramified Langlands parameters for G 2 (F ) and their L-packets. The representations appearing in these L-packets form the complete list of elliptic unipotent representations of G 2 (F ); see [CO17], where they are denoted by…”