Raising nilpotent orbits in wave-front sets

Abstract

Abstract. We study wave-front sets of representations of reductive groups over global or non-archimedean local fields.

A reciprocal branching problem for automorphic representations and global Vogan packets

“…By [26,Section I.6], nilpotent orbits corresponding to the partition [(2n+1)(2n−1)1 3 ] are parametrized by certain quadratic forms {β 2n+1 , β 2n−1 , q V 0 }, corresponding to the parts (2n+1), (2n−1) and 1 3 , where β 2n+1 and β 2n−1 are square classes, and q V 0 is the quadratic form in 3 variables on V 0 (see Section 2.1). This parametrization can be refined according to [13,Proposition 8.1], that is, E τ ⊗σ actually has a non-zero generalized Whittaker-Fourier coefficient attached to the nilpotent O, corresponding to the partition [(2n + 1)(2n − 1)1 3 ] and parametrized by quadratic forms {β, −β, q V 0 } for some β ∈ F × . Note that the normalization of the bilinear form for the irreducible representation of sl 2 (C) of dimension i in [13] differs from the one in [26] by the factor (−1) [(i−1)/2] (See [13,Section 7]).…”

“…This parametrization can be refined according to [13,Proposition 8.1], that is, E τ ⊗σ actually has a non-zero generalized Whittaker-Fourier coefficient attached to the nilpotent O, corresponding to the partition [(2n + 1)(2n − 1)1 3 ] and parametrized by quadratic forms {β, −β, q V 0 } for some β ∈ F × . Note that the normalization of the bilinear form for the irreducible representation of sl 2 (C) of dimension i in [13] differs from the one in [26] by the factor (−1) [(i−1)/2] (See [13,Section 7]). In the following, we show that such a β will suffice for the theorem.…”

“…More explicitly, first, using results in [11], we show that E τ ⊗σ has a non-zero generalized Whittaker-Fourier coefficient attached to the partition [(2n) 2 1 3 ], which is not special. Then, using results in [13], we show that E τ ⊗σ has a non-zero generalized Whittaker-Fourier coefficient attached to the partition [(2n + 1)(2n − 1)1 3 ], which is the smallest orthogonal special partition bigger than [(2n) 2 1 3 ]. This eventually implies that there exists β ∈ F × , such that D ψ n,β (E τ ⊗σ ) is non-zero.…”

“…One of them is the result in [11] on relations between degenerate Whittaker models and generalized Whittaker models of representations (see §5.1 for definitions). Another one is the result in [13] on raising nilpotent orbits in the wave front set of representations. They are indispensable for this new way to establish the global non-vanishing of the construction of the twisted automorphic descents.…”

Fourier coefficients attached to small automorphic representations of SLn(A)

“…The main goal of the present paper is to use the techniques of [JL13,JLS16,GGS17], in particular the notion of Whittaker pair, to extend the above results to all of SL n .…”